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),
with the lapse![N](article957x.gif)
![i N](article958x.gif)
![γij](article959x.gif)
![γij](article960x.gif)
![t](article961x.gif)
![(3) R](article963x.gif)
![γ](article964x.gif)
![(3) R](article965x.gif)
![kij](article966x.gif)
![γij](article968x.gif)
![γ](article969x.gif)
![γij](article971x.gif)
![ij p](article972x.gif)
This result is fully generalizable to any number of dimensions, and in spacetime dimensions,
gravitational waves carry
polarizations. We now move to the case of massive gravity.
7.1.2 ADM counting in massive gravity
We now amend the GR Lagrangian with a potential . As already explained, this can only be performed
by breaking covariance (with the exception of a cosmological constant). This potential could be a priori an
arbitrary function of the metric, but contains no derivatives and so does not affect the definition of the
conjugate momenta
This translates directly into a potential at the level of the Hamiltonian density,
If depends non-linearly on the shift or the lapse then these are no longer directly Lagrange
multipliers (if they are non-linear, they still appear at the level of the equations of motion, and so they do
not propagate a constraint for the metric but rather for themselves). As a result for an arbitrary potential
one is left with
degrees of freedom in the three-dimensional metric and its momentum conjugate
and no constraint is present to reduce the phase space. This leads to
degrees of freedom in field space:
the two usual transverse polarizations for the graviton (as we have in GR), in addition to two ‘vector’
polarizations and two ‘scalar’ polarizations.
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 spacetime dimensions, a
massive spin-2 field should propagate the same number of degrees of freedom as a massless spin-2
field in
dimensions, that is
polarizations. However, an arbitrary
potential would allow for
independent degrees of freedom, which is 1 too many
excitations, always corresponding to one BD ghost degree of freedom in an arbitrary number of
dimensions.
The only way this counting can be wrong is if the constraints for the shift and the lapse cannot be
inverted for the shift and the lapse themselves, and thus at least one of the equations of motion from the
shift or the lapse imposes a constraint on the three-dimensional metric . This loophole was first
presented in [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
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 , (where for simplicity and definiteness we take Minkowski as
the reference metric
, although most of what follows can be easily generalizable to an arbitrary
reference metric
). Expanding the lapse as
, we have
and
. In the ADM decomposition, the Fierz–Pauli mass term is then (see Eq. (2.45*))
![{γij,pij}](article996x.gif)
In Ref. [111*], the most general potential was considered up to quartic order in the , and it was
shown that there is no choice of such potential (apart from a pure cosmological constant) which would
prevent the lapse from entering non-linearly. While this result is definitely correct, it does not however
imply the absence of a constraint generated by the set of shift and lapse
. Indeed
there is no reason to believe that the lapse should necessarily be the quantity to generates the
constraint necessary to remove the BD ghost. Rather it can be any combination of the lapse and the
shift.
Example on how to evade the BD ghost non-linearly
As an instructive example presented in [137*], consider the following Hamiltonian,
with the following example for the potential 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 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 In the example (7.10*), one has 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
in terms of which the Hamiltonian is then explicitly linear in the lapse, and generates a constraint that can be read for![{ni,γij,pij}](article1006x.gif)
Condition to evade the ghost
To summarize, the condition to eliminate (at least half of) the BD ghost is that the of the
Hessian (7.13*)
vanishes as explained in [144*]. This was shown to be the case in the
ghost-free theory of massive gravity (6.3*) [(6.1*)] exactly in some cases and up to quartic order,
and then fully non-linearly in [295*]. We summarize the derivation in the general case in what
follows.
Ultimately, this means that in massive gravity we should be able to find a new shift related to the
original one as follows
, such that the Hamiltonian takes the following
factorizable form
![ni](article1012x.gif)
![i 𝒞1(γ,p)ℱ (γ,p,n (γ)) + 𝒞2(γ,p) = 0](article1013x.gif)
![i n](article1014x.gif)
Primary constraint
We now proceed by deriving the primary first-class constraint present in ghost-free (dRGT) massive gravity.
The proof works equally well for any reference at no extra cost, and so we consider a general reference
metric in its own ADM decomposition, while keep the dynamical metric
in its
original ADM form (since we work in unitary gauge, we may not simplify the metric further),
![ij p](article1018x.gif)
![γij](article1019x.gif)
![¯fij](article1020x.gif)
Proceeding similarly as in the previous example, we perform a change of variables similar as in (7.15*)
(only more complicated, but which remains linear in the lapse when expressing in terms of
) [295*, 296*]
![Di j](article1024x.gif)
The field redefinition naturally involves a square root through the expression of the matrix
in (7.21*), which should come as no surprise from the square root structure of the potential term.
For the potential to be writable in the metric language, the square root in the definition of
the tensor
should exist, which in turns imply that the square root in the definition of
in (7.21*) must also exist. While complicated, the important point to notice is that this
field redefinition remains linear in the lapse (and so does not spoil the standard constraints of
GR).
The Hamiltonian for massive gravity is then
where![𝒰](article1033x.gif)
![i 𝒰 (γ,N ,N )](article1034x.gif)
![N](article1035x.gif)
![i N](article1036x.gif)
![𝒰](article1037x.gif)
![ni](article1038x.gif)
![𝒰(γ,ni,N )](article1039x.gif)
![i n](article1040x.gif)
![β](article1043x.gif)
![α](article1044x.gif)
The structure of the potential is so that the equations of motion with respect to the shift
are independent of the lapse and impose the following relations in terms of
,
![ni](article1048x.gif)
![γij](article1049x.gif)
![pij](article1050x.gif)
![¯fij](article1051x.gif)
![𝒩¯i](article1052x.gif)
The two requirements defined previously are thus satisfied: a. The Hamiltonian is linear in the lapse and
b. the equations of motion with respect to the shift are independent of the lapse, which is
sufficient to infer the presence of a primary constraint. This primary constraint is derived by
varying with respect to the lapse and evaluating the shift on the constraint surface (7.29*),
![≈](article1055x.gif)
![𝒞0 = 0](article1056x.gif)
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,
with![H = ∫ d3xℋ mGR,1 mGR,1](article1058x.gif)
![γij](article1061x.gif)
The important point to notice is that the secondary constraint (7.33*) only depends on the phase space
variables and not on the lapse
. Thus it constraints the phase space variables rather than the
lapse and provides a genuine secondary constraint in addition to the primary one (7.30*) (indeed one can
check that
.).
Finally, we should also check that this secondary constraint is also maintained in time. This was performed [294*], by inspecting the condition
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 }](article1067x.gif)
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 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 is the equation of motion
with respect to the field
, the correct requirement for the absence of ghost is that the Hessian
defined as
7.2.2 Two-dimensional case
Consider massive gravity on a two-dimensional space-time, , with the
two Stückelberg fields
[145*]. In this case the graviton potential can only have one independent
non-trivial term, (excluding the tadpole),
![± ϕ](article1080x.gif)
As shown in Refs. [144*, 145] the square root can be traded for an auxiliary non-dynamical variable
. In this two-dimensional example, the mass term (7.43*) can be rewritten with the help of an auxiliary
non-dynamical variable
as
7.2.3 Full proof
The full proof in the minimal model (corresponding to and
and
in (6.3*)
or
in the alternative formulation (6.23*)), was derived in Ref. [297*]. We briefly review the
essence of the argument, although the full technical derivation is beyond the scope of this review and refer
the reader to Refs. [297*] and [15] for a fully-fledged derivation.
Using a set of auxiliary variables (with
, so these auxiliary variables contain ten
elements in four dimensions) as explained previously, we can rewrite the potential term in the minimal
model as [79, 342],
![Y](article1092x.gif)
![X](article1093x.gif)
![λ](article1094x.gif)
![Y](article1095x.gif)
![V](article1098x.gif)
![detV = 0](article1099x.gif)
![f](article1100x.gif)
![f](article1101x.gif)
![ϕa](article1102x.gif)
![ab λ](article1105x.gif)
![α λ](article1109x.gif)
![Cab](article1111x.gif)
![det V = 0](article1113x.gif)
![det C = 0](article1114x.gif)
![{γij,pa}](article1116x.gif)
![det f ⁄= 0](article1117x.gif)
![𝒞 1](article1118x.gif)
![πij](article1120x.gif)
![γij](article1121x.gif)
![∇ (f)](article1122x.gif)
![f](article1123x.gif)
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
where the Stückelberg background fields are given by
), the background
metric is given by
, and the background Stückelberg fields are given by
.
We now add fluctuations about that background,
with![a a a A = A bg + a](article1129x.gif)
Flat background metric
First, note that if we consider a flat background metric to start with, then at zeroth order
in , the ghost-free potential is of the form [400*], (this can also be seen from [238*, 419*])
![Fab = ∂aAb − ∂bAa](article1132x.gif)
![∂μ ϕa bg](article1133x.gif)
![F bg = 0 ab](article1134x.gif)
![a](article1135x.gif)
![U (1)](article1136x.gif)
![Aa = Aa + aa bg](article1137x.gif)
![aa](article1138x.gif)
![fμν = ∂ μaν − ∂νaμ](article1140x.gif)
![B¯μανβ](article1141x.gif)
![Aabg](article1142x.gif)
![U (1)](article1143x.gif)
![aa](article1144x.gif)
![U (1)](article1145x.gif)
Non-symmetric background Stückelberg
If the background configuration is not symmetric, then at every point one needs to perform first an internal
Lorentz transformation in the Stückelberg field space, so as to align them with the coordinate basis
and recover a symmetric configuration for the background Stückelberg fields. In this new Lorentz frame,
the Stückelberg fluctuation is
. As a result, to quadratic order in the Stückelberg
fluctuation the part of the ghost-free potential which is independent of the metric fluctuation and
its curvature goes symbolically as (7.60*) with
replaced by
, (with
). Interestingly, the Lorentz boost
now plays the role of a mass term for
what looks like a gauge field
. This mass term breaks the
symmetry, but there is still no kinetic
term for
, very much as in a Proca theory. This part of the potential is thus manifestly ghost-free (in
the sense that it provides a dynamics for only three of the four Stückelberg fields, independently of the
background).
Next, we consider the mixing with metric fluctuation while still assuming zero curvature. At linear
order in
, the ghost-free potential, (6.3*) goes as follows
![X (μnν)](article1158x.gif)
![π](article1159x.gif)
![Πμν](article1160x.gif)
![∂ μAν + ∂νA μ](article1161x.gif)
![(1,2,3) X μν](article1162x.gif)
![Aμ](article1163x.gif)
![X (μnν)](article1164x.gif)
![a μ](article1165x.gif)
![Abg μ](article1166x.gif)
![hX](article1167x.gif)
![∂0a0](article1168x.gif)
![h 00](article1169x.gif)
![h 0i](article1170x.gif)
![a0](article1171x.gif)
![hij](article1172x.gif)
![a0](article1173x.gif)
As for the second type of term in (7.61*), since on the background field
,
the second type of terms is forced to be proportional to
and cannot involve any
at all. As a result
is not dynamical, which ensures that the theory is free from the BD
ghost.
This part of the argument generalizes easily for non symmetric background Stückelberg configurations,
and the same replacement still ensures that
acquires no dynamics
from (7.61*).
Background curvature
Finally, to complete the argument, we consider the effect from background curvature, then ,
with
. The space-time curvature is another source of ‘misalignment’ between the
coordinates and the Stückelberg fields. To rectify for this misalignment, we could go two ways: Either
perform a local change of coordinate so as to align the background metric
with the flat reference
metric
(i.e., going to local inertial frame), or the other way around: i.e., express the flat reference
metric in terms of the curved background metric,
, in terms of the inverse vielbein,
. Then the building block of ghost-free massive gravity is the matrix
, defined previously
as
![a ϕ](article1189x.gif)
![∂μA ν → gbμgν − gbνgαeαa∂μϕa](article1190x.gif)
![aa](article1191x.gif)
![aa → ¯a μ = gbgeνaa μν a](article1192x.gif)
![f → ¯f + (∂Σ )Σ−1¯a](article1193x.gif)
![Σ = g˙e](article1194x.gif)
![¯a0](article1195x.gif)
![¯aμ](article1196x.gif)
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, . For simplicity we specialize further to the case
where
, so that the symmetric vielbein condition imposes
. Under this
condition, the vielbein contains as many independent components as the metric. The symmetric
veilbein condition ensures that one is able to reformulate the theory in a metric language. In
spacetime dimensions, there is a priori
independent components in the symmetric
vielbein.
Varying the action (6.1*) with respect to the vielbein leads to the modified Einstein equation,
with![Ga = 𝜀abcdωbc ∧ ed](article1203x.gif)
![𝒟Ga = dGa − ωbaGb](article1204x.gif)
![d](article1205x.gif)
![d(d − 1)∕2](article1207x.gif)
![ma](article1209x.gif)
![cn](article1210x.gif)
![c1 ⁄= 0](article1211x.gif)
![oabc = 2eaμebν∂ [μeν]c](article1214x.gif)
![eab∂[beaa] = 0](article1215x.gif)
![ma = ea](article1216x.gif)
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 -dimensional Lorentz indices,
, while lowercase Latin
indices represent the
-dimensional Lorentz indices along the space directions after ADM
decomposition,
. Greek indices represent
-dimensional spacetime indices
, while the ‘middle’ of the Latin alphabet indices
represent pure
space indices
. Finally, capital indices label the metric and span over
.
Let us start with non-interacting spin-2 fields. The theory has then
copies of coordinate
transformation invariance (the coordinate system associated with each metric can be changed separately),
as well as
copies of Lorentz invariance. At this level may, for each vielbein
,
we
may use part of the Lorentz freedom to work in the upper triangular form for the vielbein,
![γ(J)ij = e(J)a e(J)b δab i j](article1233x.gif)
![N](article1234x.gif)
![π(J)a i](article1237x.gif)
![e(J)ai](article1238x.gif)
![𝒞(J)0,i = 𝒞0,i(e(J),π(J))](article1239x.gif)
![d(d − 1)∕2](article1240x.gif)
Now rather than setting part of the Lorentz frames to be on the upper diagonal form for all the
vielbein (7.69*) we only use one Lorentz boost to set one of the vielbein in that form, say
, and ‘unboost’ the
other frames, so that for any of the other vielbein one has
![p(J)a](article1247x.gif)
We now consider arbitrary interactions between the fields of the form (6.1*),
![d ≤ N](article1250x.gif)
![A e(J) 0](article1251x.gif)
![N (J),N i(J)](article1252x.gif)
![e(J)Ai](article1253x.gif)
![e(J)A0](article1255x.gif)
![a 𝜀a1⋅⋅⋅adea(1J1) ∧ ⋅⋅⋅ ∧ e(Jdd)](article1256x.gif)
Notice that for the interactions, the terms can depend on all the
vielbeins
and all
the
‘boosts’
, (as mentioned previously, part of one Lorentz frame is set so that
and
is in the upper diagonal form). Following the procedure of [314], we can now solve
for the
remaining boosts by using
of the
shift equations of motion
![N](article1268x.gif)
![e(J)](article1275x.gif)
![N − 1](article1276x.gif)
![N](article1277x.gif)
![N (J)](article1278x.gif)
![N − 1](article1279x.gif)
![N (J)](article1280x.gif)
![J = 2,⋅⋅⋅,N](article1281x.gif)
![(N − 1)](article1282x.gif)
![N (1)](article1283x.gif)
![i N(1)](article1284x.gif)
We now have all the ingredients to count the number of dofs in phase space: We start with
components in each of the
vielbein
and associated conjugate momenta,
that is a total of
phase space variables. We then have
constraints21
associated with the
. There is one copy of diffeomorphism removing
phase space dofs
(with Lagrange multiplier
and
) and
additional first-class constraints
with Lagrange multipliers
removing
dofs. As a result we end up with
![(d + 1)](article1300x.gif)