Before jumping into the subtleties of massive spin-2 field and gravity in general, we start this review with massless and massive spin-1 fields as a warm up. Consider a Lorentz vector field living on a four-dimensional Minkowski manifold. We focus this discussion to four dimensions and the extension to dimensions is straightforward. Restricting ourselves to Lorentz invariant and local actions for now, the kinetic term can be decomposed into three possible contributions:
We now proceed to establish the behavior of the different degrees of freedom (dofs) present in this theory. A priori, a Lorentz vector field in four dimensions could have up to four dofs, which we can split as a transverse contribution satisfying bearing a priori three dofs and a longitudinal mode with .
Focusing on the longitudinal (or helicity-0) mode , the kinetic term takes the form, two dofs are actually hidden in with an opposite sign kinetic term. This can be seen by expressing the propagator as the sum of two propagators with opposite signs: 2.1*) with and is always sick. Alternatively, one can see the appearance of the Ostrogradsky instability by introducing a Lagrange multiplier , so that the kinetic action (2.5*) for is equivalent to 2 . We can now perform the change of variables and giving the resulting Lagrangian for the two scalar fields 3 The only way to prevent this generic pathology is to make the specific choice , which corresponds to the well-known Maxwell kinetic term.
Now that the form of the local and covariant kinetic term has been uniquely established by the requirement that no ghost rides on top of the helicity-0 mode, we focus on the remaining transverse mode ,2.11*) allows us to fix the gauge of our choice. For convenience, we perform a -split and choose Coulomb gauge , so that only two dofs are present in , i.e., contains no longitudinal mode, , with and the Coulomb gauge sets the longitudinal mode . The time-component does not exhibit a kinetic term,
Starting with the Maxwell action, we consider a covariant mass term corresponding to the Proca action2.11*), so that the Coulomb gauge can no longer be chosen and the longitudinal mode is now dynamical. To see this, let us use the previous decomposition and notice that the mass term now introduces a kinetic term for the helicity-0 mode ,
Before moving to the Abelian Higgs mechanism, which provides a dynamical way to give a mass to bosons, we first comment on the discontinuity in number of dofs between the massive and massless case. When considering the Proca action (2.16*) with the properly normalized fields and , one does not recover the massless Maxwell action (2.9*) or (2.10*) when sending the boson mass . A priori, this seems to signal the presence of a discontinuity which would allow us to distinguish between for instance a massless photon and a massive one no matter how tiny the mass. In practice, however, the difference is physically indistinguishable so long as the photon couples to external sources in a way which respects the symmetry. Note however that quantum anomalies remain sensitive to the mass of the field so the discontinuity is still present at this level, see Refs. [197, 204].
To physically tell the difference between a massless vector field and a massive one with tiny mass, one has to probe the system, or in other words include interactions with external sources
Notice that in the massive case no symmetry is present and the source needs not be conserved. However, the previous argument remains unchanged so long as goes to zero in the massless limit at least as quickly as the mass itself. If this condition is violated, then the helicity-0 mode ought to be included in the exchange amplitude (2.21*). In parallel, in the massless case the non-conserved source provides a new kinetic term for the longitudinal mode which then becomes dynamical.
Associated with the absence of an intrinsic discontinuity in the massless limit is the existence of a Higgs mechanism for the vector field whereby the vector field acquires a mass dynamically. As we shall see later, the situation is different for gravity where no equivalent dynamical Higgs mechanism has been discovered to date. Nevertheless, the tools used to describe the Abelian Higgs mechanism and in particular the introduction of a Stückelberg field will prove useful in the gravitational case as well.
To describe the Abelian Higgs mechanism, we start with a vector field with associated Maxwell tensor and a complex scalar field with quartic potential2.11*) shifts the scalar field as
Now that we have introduced the notion of a massless and a massive spin-1 field, let us look at interacting spin-1 fields. We start with free and massless gauge fields, , with , and respective Maxwell tensors ,
However, in addition to these gauge invariances, the kinetic term is invariant under global rotations in field space,
We can see this statement more explicitly in the case of spin-1 fields by diagonalizing the mass matrix . A mentioned previously, the kinetic term is invariant under field space rotations, (2.29*), so one can use this freedom to work in a field representation where the mass matrix is diagonal,
As we have seen in the case of a vector field, as long as it is local and Lorentz-invariant, the kinetic term is uniquely fixed by the requirement that no ghost be present. Moving now to a spin-2 field, the same argument applies exactly and the Einstein–Hilbert term appears naturally as the unique kinetic term free of any ghost-like instability. This is possible thanks to a symmetry which projects out all unwanted dofs, namely diffeomorphism invariance (linear diffs at the linearized level, and non-linear diffs/general covariance at the non-linear level).
We consider a symmetric Lorentz tensor field . The kinetic term can be decomposed into four possible local contributions (assuming Lorentz invariance and ignoring terms which are equivalent upon integration by parts):2.32*) with untuned coefficients would contain higher derivatives for which in turn would imply a ghost. As we shall see below, avoiding a ghost within the kinetic term automatically leads to gauge-invariance. After substitution of in terms of and , the potentially dangerous parts are
In spacetime dimensions, gravitational waves have independent polarizations. This means that in three dimensions there are no gravitational waves and in five dimensions they have five independent polarizations.
As seen in seen in Section 2.2.1, for a local and Lorentz-invariant theory, the linearized kinetic term is uniquely fixed by the requirement that longitudinal modes propagate no ghost, which in turn prevents that operator from exciting these modes altogether. Just as in the case of a massive spin-1 field, we shall see in what follows that the longitudinal modes can nevertheless be excited when including a mass term. In what follows we restrict ourselves to linear considerations and spare any non-linearity discussions for Parts I and II. See also  for an analysis of the linearized Fierz–Pauli theory using Bardeen variables.
In the case of a spin-2 field , we are a priori free to choose between two possible mass terms and , so that the generic mass term can be written as a combination of both,. Including the four linearized Stückelberg fields, the resulting mass term 2.1*) in Section 2.1.1 with and . Now the same logic as in Section 2.1.1 applies and singling out the longitudinal component of these Stückelberg fields it follows that the only combination which does not involve higher derivatives is or in other words . As a result, the only possible mass term one can consider which is free from an Ostrogradsky instability is the Fierz–Pauli mass term unitary gauge, i.e., in the gauge where the Stückelberg fields are set to zero, the Fierz–Pauli mass term simply reduces to
To identify the propagating degrees of freedom we may split further into a transverse and a longitudinal mode,
In terms of and the Stückelberg fields and the linearized Fierz–Pauli action is
Terms on the first line represent the kinetic terms for the different fields while the second line represent the mass terms and mixing.
We see that the kinetic term for the field is hidden in the mixing with . To make the field content explicit, we may diagonalize this mixing by shifting and the linearized Fierz–Pauli action is
The degrees of freedom have not yet been split into their mass eigenstates but on doing so one can easily check that all the degrees of freedom have the same positive mass square .
Most of the phenomenology and theoretical consistency of massive gravity is related to the dynamics of the helicity-0 mode. The coupling to matter occurs via the coupling , where is the trace of the external stress-energy tensor. We see that the helicity-0 mode couples directly to conserved sources (unlike in the case of the Proca field) but the helicity-1 mode does not. In most of what follows we will thus be able to ignore the helicity-1 mode.
As we shall see in Section 9.1, the graviton mass can also be promoted to a scalar function of one or many other fields (for instance of a different scalar field), . We can thus wonder whether a dynamical Higgs mechanism for gravity can be considered where the field(s) start in a phase for which the graviton mass vanishes, and dynamically evolves to acquire a non-vanishing vev for which . Following the same logic as the Abelian Higgs for electromagnetism, this strategy can only work if the number of dofs in the massless phase is the same as that in the massive case . Simply promoting the mass to a function of an external field is thus not sufficient since the graviton helicity-0 and -1 modes would otherwise be infinitely strongly coupled as .
To date no candidate has been proposed for which the graviton mass could dynamically evolve from a vanishing value to a finite one without falling into such strong coupling issues. This does not imply that Higgs mechanism for gravity does not exist, but as yet has not been found. For instance on AdS, there could be a Higgs mechanism as proposed in , where the mass term comes from integrating out some conformal fields with slightly unusual (but not unphysical) ‘transparent’ boundary conditions. This mechanism is specific to AdS and to the existence of time-like boundary and would not apply on Minkowski or dS.
As in the case of spin-1, the massive spin-2 field propagates more dofs than the massless one. Nevertheless, these new excitations bear no observational signatures for the spin-1 field when considering an arbitrarily small mass, as seen in Section 2.1.2. The main reason for that is that the helicity-0 polarization of the photon couple only to the divergence of external sources which vanishes for conserved sources. As a result no external sources directly excite the helicity-0 mode of a massive spin-1 field. For the spin-2 field, on the other hand, the situation is different as the helicity-0 mode can now couple to the trace of the stress-energy tensor and so generic sources will excite not only the 2 helicity-2 polarization of the graviton but also a third helicity-0 polarization, which could in principle have dramatic consequences. To see this more explicitly, let us compute the gravitational exchange amplitude between two sources and in both the massive and massless gravitational cases.
In the massless case, the theory is diffeomorphism invariant. When considering coupling to external sources, of the form , we thus need to ensure that the symmetry be preserved, which implies that the stress-energy tensor should be conserved . When computing the gravitational exchange amplitude between two sources we thus restrict ourselves to conserved ones. In the massive case, there is a priori no reasons to restrict ourselves to conserved sources, so long as their divergences cancel in the massless limit .
Let us start with the massive case, and consider the response to a conserved external source ,[465*, 497*]. This linearized vDVZ discontinuity was recently repointed out in .) As has been known for many decades, this discontinuity (or the fact that the Ricci scalar vanishes) is an artefact of the linearized theory and is resolved by the Vainshtein mechanism [463*] as we shall see later.
Plugging these expressions back into the modified Einstein equation, we get
2.38*), one can choose a gauge of our choice, for instance de Donder (or harmonic) gauge 2.60*) in the small mass limit. This difference between the massless limit of the massive propagator and the massless propagator (and gravitational exchange amplitude) is a well-known fact and was first pointed out by van Dam, Veltman and Zakharov in 1970 [465, 497]. The resolution to this ‘problem’ lies within the Vainshtein mechanism . In 1972, Vainshtein showed that a theory of massive gravity becomes strongly coupled a low energy scale when the graviton mass is small. As a result, the linear theory is no longer appropriate to describe the theory in the limit of small mass and one should keep track of the non-linear interactions (very much as what we do when approaching the Schwarzschild radius in GR.) We shall see in Section 10.1 how a special set of interactions dominate in the massless limit and are responsible for the screening of the extra degrees of freedom present in massive gravity.
Another ‘non-GR’ effect was also recently pointed out in Ref.  where a linear analysis showed that massive gravity predicts different spin-orientations for spinning objects.
When considering the massless and non-interactive spin-2 field in Section 2.2.1, the linear gauge invariance (2.38*) is exact. However, if this field is to be probed and communicates with the rest of the world, the gauge symmetry is forced to include non-linear terms which in turn forces the kinetic term to become fully non-linear. The result is the well-known fully covariant Einstein–Hilbert term , where is the scalar curvature associated with the metric .
To see this explicitly, let us start with the linearized theory and couple it to an external source , via the coupling2.68*) fails to be conserved. When considering the coupling (2.67*), the Klein–Gordon equation receives corrections of the order of 4), which requires the stress-energy tensor to be covariantly conserved. To satisfy this symmetry, the kinetic term (2.36*) should then be promoted to a fully non-linear contribution, 5 As a result, any theory of an interacting spin-2 field is necessarily fully non-linear and leads to the theory of gravity where non-linear diffeomorphism invariance (or covariance) plays the role of the local gauge symmetry that projects out four out of the potential six degrees of freedom of the graviton and prevents the excitation of any ghost by the kinetic term.
The situation is very different from that of a spin-1 field as seen earlier, where coupling with other fields can be implemented at the linear order without affecting the gauge symmetry. The difference is that in the case of a symmetry, there is a unique nonlinear completion of that symmetry, i.e., the unique nonlinear completion of a is nothing else but a . Thus any nonlinear Lagrangian which preserves the full symmetry will be a consistent interacting theory. On the other hand, for spin-2 fields, there are two, and only two ways to nonlinearly complete linear diffs, one as linear diffs in the full theory and the other as full non-linear diffs. While it is possible to write self-interactions which preserve linear diffs, there are no interactions between matter and which preserve linear diffs. Thus any theory of gravity must exhibit full nonlinear diffs and is in this sense what leads us to GR.
We have introduced the spin-2 field as the perturbation about flat spacetime. When considering the theory of a field of given spin it is only natural to work with Minkowski as our spacetime metric, since the notion of spin follows from that of Poincaré invariance. Now when extending the theory non-linearly, we may also extend the theory about different reference metric. When dealing with a reference metric different than Minkowski, one loses the interpretation of the field as massive spin-2, but one can still get a consistent theory. One could also wonder whether it is possible to write a theory of massive gravity without the use of a reference metric at all. This interesting question was investigated in [75*], where it shown that the only consistent alternative is to consider a function of the metric determinant. However, as shown in [75*], the consistent function of the determinant is the cosmological constant and does not provide a mass for the graviton.
Full diffeomorphism invariance (or covariance) indicates that the theory should be built out of scalar objects constructed out of the metric and other tensors. However, as explained previously a theory of massive gravity requires the notion of a reference metric6 (which may be Minkowski ) and at the linearized level, the mass for gravity was not built out of the full metric , but rather out of the fluctuation about this reference metric which does not transform as a tensor under general coordinate transformations. As a result the mass term breaks covariance.
This result is already transparent at the linear level where the mass term (2.39*) breaks linearized diffeomorphism invariance. Nevertheless, that gauge symmetry can always be ‘formally’ restored using the Stückelberg trick which amounts to replacing the reference metric (so far we have been working with the flat Minkowski metric as the reference), to
Following the same Stückelberg trick non-linearly, one can ‘formally restore’ covariance by including four Stückelberg fields () and promoting the reference metric , which may of may not be Minkowski, to a tensor [446*, 27*],unitary gauge, where the Stückelberg fields are , we simply recover .
This Stückelberg trick for massive gravity dates already from Green and Thorn  and from Siegel , introduced then within the context of open string theory. In the same way as the massless graviton naturally emerges in the closed string sector, open strings also have spin-2 excitations but whose lowest energy state is massive at tree level (they only become massless once quantum corrections are considered). Thus at the classical level, open strings contain a description of massive excitations of a spin-2 field, where gauge invariance is restored thanks to same Stückelberg fields as introduced in this section. In open string theory, these Stückelberg fields naturally arise from the ghost coordinates. When constructing the non-linear theory of massive gravity from extra dimension, we shall see that in that context the Stückelberg fields naturally arise at the shift from the extra dimension.
For later convenience, it will be useful to construct the following tensor quantity,
An alternative way to Stückelberize the reference metric is to express it as[14*], both matrices and have the same eigenvalues, so one can choose either one of them in the definition of the massive gravity Lagrangian without any distinction. The formulation in terms of rather than was originally used in Ref. , although unsuccessfully as the potential proposed there exhibits the BD ghost instability, (see for instance Ref. ).
If we now focus on the flat reference metric, , we may further split the Stückelberg fields as and identify the index with a Lorentz index,7 we obtain the non-linear generalization of the Stückelberg trick used in Section 188.8.131.52*) of in terms of the helicity-0 and -1 modes and all indices are raised and lowered with respect to .
In other words, the fluctuations about flat spacetime are promoted to the tensor[143*].
The most straightforward non-linear extension of the Fierz–Pauli mass term is as follows. Alternatively, one may also generalize the Fierz–Pauli mass non-linearly as follows [75*] II, we shall see that the extension of the Fierz–Pauli to a non-linear theory free of the BD ghost is unique (up to two constant parameters).
The easiest way to see the appearance of a ghost at the non-linear level is to follow the Stückelberg trick non-linearly and observe the appearance of an Ostrogradsky instability [111*, 173*], although the original formulation was performed in unitary gauge in [75*] in the ADM language (Arnowitt, Deser and Misner, see Ref. ). In this section we shall focus on the flat reference metric, .2.83*) reads 2.86*) is a total derivative, which is another way to see the special structure of the Fierz–Pauli mass term. Unfortunately this special fact does not propagate to higher order and the cubic and quartic interactions are genuine higher order operators which lead to equations of motion with quartic and cubic derivatives. In other words these higher order operators and propagate an additional degree of freedom which by Ostrogradsky’s theorem, always enters as a ghost. While at the linear level, these operators might be irrelevant, their existence implies that one can always find an appropriate background configuration , such that the ghost is manifest 2.84*).
Alternatively the mass term was also generalized to include curvature invariants as in Ref. . This theory was shown to be ghost-free at the linear level on FLRW but not yet non-linearly.
Instead to prevent the presence of the BD ghost fully non-linearly (or equivalently about any background), one should construct the mass term (or rather potential term) in such a way, that all the higher derivative operators involving the helicity-0 mode are total derivatives. This is precisely what is achieved in the “ghost-free” model of massive gravity presented in Part II. In the next Part I we shall use higher dimensional GR to get some insight and intuition on how to construct a consistent theory of massive gravity.