Recombinant DNA technology (Immunological screening)
Part III Essay: Could the graviton have a mass?
1. Part III Essay
Could the graviton have a mass?
Abstract
General relativity is the unique theory of a massless spin 2 particle,
called the graviton. Massive gravity theories are field theories of a massive
spin 2 particle. One cannot uniquely extend general relativity into massive
gravity and in general such extensions suffer from pathologies such as
discontinuity with general relativity in the massless limit, the presence
of ghost fields and lack of renormalizability. The goal of this essay is
to shed light on the problems involved in giving the graviton a mass
and explain how they might be resolved. In particular, we discuss two
proposed methods to deal with the problems, the Vainshtein mechanism
and dRGT construction.
Yiteng Dang
April 2014
3. 1 Introduction
In quantum field theory one typically discusses theories of scalar bosons (spin
0), fermions (spin 1/2) and vector bosons (spin 1). The free action for these
theories are
Sφ = dD
x ∂µφ∂µ
φ + m2
φ2
(1)
Sψ = dD
x ¯ψ(i/∂ + m)ψ (2)
SA = dD
x −
1
4
FµνFµν
, (3)
where Fµν = ∂µAν −∂νAµ is the field strength tensor. The compactness of their
form leads to elegant equations of motion. On the other hand, the easiest form
of an action describing a free massive spin 2 particle takes the form
SF P = M2
P dD
x −
1
2
∂λhµν∂λ
hµν
+ ∂µhνλ∂ν
hµλ
− ∂µhµν
∂νh
+
1
2
∂λh∂λ
h −
1
2
m2
(hµνhµν
− h2
) , (4)
where hµν is symmetric and MP is the Planck mass. The equations of motion
resulting from this action are substantially more complicated, hinting at the dif-
ficulty of constructing a field theory for massive gravity. This above action was
first proposed by Fierz and Pauli in 1939 and has been studied intensively since
then. It constitutes the simplest case of a family of theories for a massive spin 2
particle. We will refer this theory, which may be extended to include coupling
to matter, as linear Fierz-Pauli (FP) theory, because the field equations arising
from them are linear. This theory turns out to suffer from a major problem: it
is not continuous with general relativity (GR) in the massless limit. The dis-
continuity is known as the van Dam, Veltman, Sacharov (vDVZ) discontinuity.
This discovery of this discontinuity has led to many attempts to extend theories
to include non-linear terms.
Why should we study a theory of a massive spin 2 particle? To begin with,
let us try to understand why GR is the theory of a massless spin 2 particle. This
explanation is largely based on Feynman’s Lectures of Gravitation [Feyn95]. In
any quantum field theory of point particles, the force between two particles with
a charge associated to that force is mediated by virtual particles. In a quantum
field theory of gravity, the particle mediating the gravitational force is called the
graviton. One can show that this particle cannot arise from existing theories
such as QED, QCD and the electroweak theory. Hence it must represent a new
field, and by ruling out other possibilities one can also show that this new field
must have spin 2. The argument is as follows: exchange particles can only be
bosons, and odd spin carriers have the property that its like charges repel (an
example is the electromagnetic force, mediated by a spin 1 photon). Spin 0
fields can only couple to the trace of the energy momentum tensor. Hence the
only possibilities left are spin 2, 4, 6, . . . , and it turns out that only considering
spin 2 is sufficient.
2
4. Furthermore, in the nonrelativistic limit of GR we recover Newtonian grav-
ity, which is described by the 1/r2
force given by Newton’s Law of Gravitation.
This is a long range force since it can be felt at arbitrary distances. Such a
force must be mediated by a massless particle, and therefore in GR the graviton
should be massless. This is in contrast with for instance electroweak theory, in
which the weak nuclear force has finite range and is mediated by massive spin
1 bosons.
Spin 0 particles are described by scalar fields, spin 1 particles by vector
fields, so it is natural to think that spin 2 particles should be described by rank
2 tensor fields. There is a subtlety here, because in addition the tensor has
to be symmetric. An anti-symmetric tensor would lead to a theory resembling
electromagnetism, which can be described in terms of the anti-symmetric field
strength tensor Fµν mentioned above.
From quantum mechanics we know that for any given spin j, there are 2j +1
helicity states with helicity h = −j, −j + 1, . . . , j − 1, j. However, the spin of a
massless particle must be aligned in the direction of its momentum, and there-
fore a massless particle can only have two helicities h = −j, j 1
. Hence we
find that the massless graviton has two polarization states, while the massive
graviton has five polarization states. This can be verified for the linear theory
by counting degrees of freedom in the Hamiltonian formalism (cf. section 2.1
in [Hint11]). However, in non-linear theories we find for a large class of such
theories an extra degree of freedom, which in certain situations corresponds to
a ghost field, a field with an opposite sign for its kinetic term.
After writing down a possible action for a massive gravity theory, one can
hope to quantize this theory and construct a quantum theory of gravity. How-
ever, one faces a serious problem: the theory might be non-renormalisable. This
is already the case for GR, which we can show by a dimensional analysis ar-
gument. The only coupling involved in the theory is the gravitational constant
G. Newtonian gravity has a potential of the form V (r) = GM1M2
r , from which
we deduce that G has mass dimension -2. Schematically, a scattering ampli-
tude for the graviton-graviton scattering process at energy E takes the form
∼ [1 + GE2
+ (GE2
)2
+ . . .] (cf. III.2 in [Zee10]). At energy E ∼ G− 1
2 , the
corrections become of order one and the amplitude diverges.
We have now identified three main problems in constructing a theory of
massive gravity. The vDVZ discontinuity violates the principle that physics
should be continuous in its parameters. However, it turns out to be inherent
to the linear theory on flat spacetime. It disappears in curved spacetime (cf.
chapter 5 in [Hint11]), while in the nonlinear theory the problem can be re-
solved by a trick known as the Vainshtein mechanism. Secondly, a ghost field
is problematic and needs to be resolved. Any positive energy scalar field leads
to an attractive force, but the force mediated by a ghost field is repulsive. This
follows from the fact that the propagator for the ghost has a different sign in
the denominator. Moreover, this sign difference in the propagator may lead
1It seems very hard to find a short but precise argument why this must be true. For the
photon it comes from the fact that in electromagnetism A0 is non-dynamical and enforces
another (gauge) constraint, but in general one would need group theoretic arguments from
chapter 2 of [Wein95]
3
5. to either theories with no probabilistic interpretation or to divergent scattering
amplitudes [Wang13], showing that the presence of a ghost leads to unphysical
phenomena. Fortunately, the ghost problem can be resolved by restricting to a
specific class of NLFP theories, as was discovered by de Rham, Gabadadze and
Tolley. This has become known as the dRGT construction, and in conjunction
with the Vainshtein mechanism it gives a ghost-free theory which does not have
the vDVZ discontinuity. The third issue is the problem of renormalisation. Al-
though gravity is non-renormalizable, one can hope to work with an effective
field theory (EFT) description. However, since such a theory is only valid at
low energy scales, it is necessary to have a sufficiently high cutoff scale2
beyond
which the theory ceases to make sense. We shall see that a low cut-off scale
can lead to serious problems, related to unwanted quantum effects entering the
stage before the Vainshtein mechanism can do its work. However, it turns out
that the dRGT construction raises the cutoff scale to a sufficiently high value
for this problem not to occur.
There are many reasons for studying massive gravity. Firstly, massive grav-
ity could shed light on the open questions in cosmology and gravity such as the
cosmological constant problem, the hierarchy problem and the reason behind the
accelerating expansion of the universe. Secondly, a consistent theory of massive
gravity would be a contestant to GR. In the case of multiple contesting theories,
ultimately experiments would need to determine which is the correct one. Even
such experiments would rule out massive gravity turns as a feasible theory of
the universe, this could be seen as strengthening support for the validity of GR.
Finally, modifying a theory requires more than superficial understanding of the
theory and may lead to interesting theoretical developments.
The outline of the essay is as follows: we first discuss linear FP theory and
its solutions, before giving a short discussion of fully non-linear massive gravity
theories. Section 3 discusses the three problems of massive gravity mentioned
above. These will each be dealt with in relatively much detail, and we will try
to point out exactly how they arise. This is followed by a discussion of the
Vainshtein mechanism and dRGT construction, techniques which are effective
in solving some of the problems discussed in the previous sections. However,
there remain novel and outstanding problems which we discuss in section 5.
Most of the theoretical material for sections 2 and 3 is based on the text by
Hinterbichler [Hint11], while the discussion of the Vainshtein mechanism and
dRGT construction in chapter 4 rely heavily on [BabDef13] and [Bab13]. I
have also made use of the more recent text by de Rham [dR14] for insights
not presented elsewhere, and the Feynman Lectures on Gravitation [Feyn95]
for understanding the general framework better. Unfortunately I have had to
limit discussing cosmological solutions as presented in [FGLM13] to a very brief
summary, due to my limited knowledge of cosmology and time constraints. I
have also consulted numerous other articles to understand certain details, as
well as textbooks and lecture notes such as [Zee10] and [Reall13].
2Throughout the essay, we shall refer to this scale as the cutoff scale, although it is more
accurate to call it the strong coupling scale, as we shall argue in section 3.2.
4
6. 2 Massive gravity theories
In this section we will give an overview of the massive gravity theories relevant to
this essay. The starting point is the linear Fierz-Pauli theory, which is obtained
by adding a mass term to the action describing linearized GR. In section 2.2
we study the linear Fierz-Pauli theory in detail, explain how it can be derived
and how a general solution can be obtained. The linear theory can be extended
to NLMG theories by adding a mass term to the full GR action rather than
the linearized action. A wide range of theories fall within this class, since the
extension of GR in the fully non-linear theory is not unique, and we shall give
the most general form of such an extension.
2.1 Linearized GR
We begin by giving a short summary of linearized GR, which will be relevant
to the discussion of linear FP theory. First recall that GR in the absence of
sources is described by the Einstein-Hilbert (EH) action
SEH = dD
x
√
−gR. (5)
Varying the EH action gives the Einstein equation
Gµν = 8πGTµν. (6)
In linearized GR we take spacetime to be R1,3
and expand the metric gµν around
flat space as gµν = ηµν +hµν. The field hµν is called the metric perturbation and
transforms as a tensor under Lorentz transformations. The Riemann curvature
tensor, Ricci tensor, curvature scalar and Einstein tensor can be expanded in
hµν. We denote the trace by h ≡ hµ
µ. To linear order, the Einstein tensor is
∂λ
∂(µhν)λ −
1
2
hµν −
1
2
∂µ∂νh −
1
2
ηµν (∂ρ
∂σ
hρσ − h) . (7)
Here denotes the d’Alembertian operator, ≡ ∂µ
∂µ. The Einstein equation
expanded to linear order hence becomes
∂λ
∂(µhν)λ −
1
2
hµν −
1
2
∂µ∂νh −
1
2
ηµν (∂ρ
∂σ
hρσ − h) = 8πGTµν. (8)
The diffeomorphism invariance of GR implies there is a gauge symmetry for the
metric perturbation which takes the form
hµν → hµν + ∂µξν + ∂νξµ. (9)
We can choose the gauge to be what is variably known as the Lorenz, de Donder
and harmonic gauge,
∂µ
¯hµν
= 0, (10)
where ¯hµν = hµν −1
2 hηµν is the trace-reversed metric perturbation. In particular
we have ¯h = −h. In this gauge, the linearized Einstein equation simplifies to
¯hµν
= −16πGTµν
. (11)
5
7. 2.2 Linear Fierz-Pauli theory
Let us now turn to linear Fierz-Pauli (FP) theory. The action for this theory is
given by
SF P = dD
x −
1
2
∂λhµν∂λ
hµν
+ ∂µhνλ∂ν
hµλ
− ∂µhµν
∂νh +
1
2
∂λh∂λ
h
−
1
2
m2
(hµνhµν
− h2
) + κhµνTµν
, (12)
where κ is a yet undermined coupling constant (its explicit value shall not be
of importance to us)3
.
The claim is that for m = 0, the action precisely describes linearized GR.
There are several ways to see this. For example, one can start with the EH
action and expand it to quadratic order. This will give precisely the massless
part of (12) (cf. chapter 6.1 in [Hint11]). Furthermore, varying the part of the
action obtained by setting m = Tµν = 0 in (12) gives a term proportional to the
linearization of the Einstein tensor (7). Hence we obtain the linearized Einstein
equation (11) by choosing an appropriate value for κ. The expressions we obtain
in this process are rather complicated, but can be simplified by making use of
gauge symmetry. One can check that for m = 0, and under the assumption that
the energy-momentum tensor is conserved (∂µ
Tµν = 0), the action has precisely
the same gauge symmetry as linearized GR, as described by (9). If we choose
the gauge to be (10), it can be shown that the equations of motion simplify to
(11). As an aside, we note that the conservation condition ∂µ
Tµν = 0 in fact
does not need to be imposed, but follows from the equations of motion for the
action (12) (cf. chapter 3 in [Hint11]).
Notice that the action contains all possible Lorentz-invariant terms involving
two fields and two derivatives. Actually this gives another way to determine the
action. Consider first the theory of a massless spin 2 field. Since the field has to
be dynamical, for the kinetic part we need quadratic terms with two derivatives.
The most general form for the kinetic term is then a linear combination of all
possible Lorentz contracted terms with two fields and two derivatives. Up to
equivalence by partial integration, the Lagrangian must be of the form
Lkin = A ∂λhµν∂λ
hµν
+ B ∂µhνλ∂ν
hµλ
+ C ∂µhµν
∂νh + D ∂λh∂λ
h (13)
where A, B, C, D are arbitrary coefficients. We can then add the only possible
matter coupling term ∼ hµνTµν
and impose that the energy-momentum tensor
Tµν
is conserved. From the equations of motion and this conservation property
it can be shown that the coefficients A, B, C, D must be precisely those in (4)
(cf. lecture 3 in [Feyn95]). For the mass term, the only Lorentz-invariant terms
we can add are proportional to hµνhµν
and h2
. The relative coefficient of −1
between them is called the Fierz-Pauli tuning and does not follow from any sym-
metry. However, both the coefficients of the kinetic term and the Fierz-Pauli
tuning follow from the requirement that the theory is ghost free (cf. section 2.2
3Note that the action differs from the previous action (4) by an overall factor of M2
P .
Wherever the energy or length scale is not of importance in the discussion, we shall set
MP = 1.
6
8. in [dR14]).
Now consider the action (12) without sources, i.e. set Tµν = 0. Let us study
its equations of motion and solve them explicitly. We can rearrange terms by
partial integration and factor out terms of hµν
to get
SF P =
1
2
dD
x hµν
hµν − ∂µ∂λhλ
ν − ∂ν∂λhλ
µ + ηµν∂ρ∂σhρσ
+∂µ∂νh − ηµν h) − m2
hµν
(hµν − ηµνh) . (14)
Note that the kinetic part is invariant if we perform partial integration twice.
Hence we can simply read off the equations of motion:
hµν − ∂µ∂λhλ
ν − ∂ν∂λhλ
µ + ηµν∂ρ∂σhρσ
+ ∂µ∂νh − ηµν h
− m2
(hµν − ηµνh) = 0. (15)
In this form it is hard to see how we might obtain an explicit solution for hµν.
The loss of the gauge symmetry means that we cannot further simplify this
expression by choosing an appropriate gauge. However, one can show that (15)
is equivalent to a set of three equations. To see this, first act with ∂µ
on (15)
to get
∂µ
hµν − ∂λhλ
ν − ∂µ
∂ν∂λhλ
µ + ∂ν∂ρ∂σhρσ
+ ∂νh − ∂ν h
− m2
(∂µ
hµν − ∂νh)
= −m2
(∂µ
hµν − ∂νh) = 0. (16)
If m2
= 0, this gives the constraint ∂µ
hµν − ∂νh = 0. Plugging this back into
(15) gives
hµν − ∂µ∂νh − ∂ν∂µh + ηµν h + ∂µ∂νh − ηµν h − m2
(hµν − ηµνh)
= hµν − ∂µ∂νh − m2
(hµν − ηµνh) = 0. (17)
Taking the trace of this expression yields
h − h − m2
(1 − D)h = m2
(1 − D)h = 0, (18)
giving the constraint h = 0. Hence the previous constraint reduces to ∂µ
hµν = 0.
Plugging both constraints into (15) gives ( −m2
)hµν = 0, since all other terms
contain either h or ∂µ
hµν. Thus, we have shown that (15) is equivalent to the
set of three equations
( − m2
)hµν = 0
∂µ
hµν = 0
h = 0. (19)
We recognize the first equation as the Klein-Gordon equation, while the second
and third equations can be interpreted as constraints. The solution of the Klein-
Gordon equation takes the general form
hµν
(x) =
dd
p
(2π)d2ωp
hµν
(p)eip·x
+ hµν∗
(p)e−ip·x
. (20)
7
9. Here d = D −1 denotes the number of spatial dimensions, p denotes the spatial
momenta, pµ
= (ωp, p) with ωp = p2 + m2, and hµν
(p) are Fourier coeffi-
cients.
We can expand the Fourier coefficients in a basis of symmetric tensors µν
(p, λ),
indexed by λ, and write
hµν
(p) = ap,λ
µν
(p, λ). (21)
At this point the ap,λ are simply coefficients of the basis tensors, but if we
would quantize the theory they would take the role of creation and annihilation
operators for graviton states.
More explicitly, the basis can be chosen by specifying tensors µν
(k, λ) for a
fixed momentum kµ
, and imposing that it transforms under usual tensor trans-
formation rules under Lorentz transformations. The constraints from (19) then
translate to constraints on µν
(k, λ), and we find
kµ
µν
(k, λ) = 0, ηµν
µν
(k, λ) = 0. (22)
Let kµ be (m, 0, . . . , 0). Then the first condition tells that µν
(k, λ) must be
purely spatial, while the second implies that its trace vanishes. Hence together
they imply that µν
(k, λ) is a symmetric traceless spatial tensor and therefore
λ = 1, . . . , d(d+1)
2 . Similar conditions hold for µν
(p, λ) upon boosting to mo-
mentum pµ
.
We can now write the general solution as
hµν
(x) =
dd
p
(2π)d2ωp λ
ap,λ
µν
(p, λ)eip·x
+ a∗
p,λ
µν∗
(p, λ)e−ip·x
. (23)
Define the mode functions
uµν
p,λ(x) ≡
1
(2π)d2ωp
µν
(p, λ)eip·x
, (24)
so that the solution (23) simplifies to
hµν
(x) = dd
p
λ
ap,λuµν
p,λ(x) + a∗
p,λuµν∗
p,λ (x) . (25)
At this point, we could perform canonical quantisation by imposing commu-
tation relations (since gravitons are bosons) on ap,λ and a∗
p,λ and promoting
them to operators creating graviton states. We would also need the canonical
momenta in this procedure, which we have not derived but are easy to calculate
(cf. section 2.1 in [Hint11]). However, this would be an attempt to construct a
quantum theory of gravity and falls beyond the scope of the essay. Throughout
most of the rest of the essay it shall be sufficient to deal with the classical field
theory.
2.3 Non-linear massive gravity
Instead of taking linearized GR as the starting point and adding mass terms
to its action, we can also start by considering the full GR action, described
8
10. by the EH action (5). Recall that this action is invariant under spacetime
diffeomorphisms of the form
x → f(x), gµν
(x) → ∂fα
∂xµ
∂fβ
∂xν gαβ
(f(x)). (26)
Adding a mass term breaks this gauge invariance. Hence just as in linear massive
gravity this means that the precise form of the mass term cannot be determined
from some symmetry property. However, we can impose the requirement that
under the replacement gµν = ηµν +hµν, expanding the total action to quadratic
order in hµν should give the FP action (12)4
. Since the Einstein-Hilbert action
expanded to quadratic order gives the massless part of the FP, this requirement
means that the mass term expanded to quadratic order should give the mass
term in the FP action.
In general, we do not need to expand around the flat metric but can make
any expansion gµν = fµν +hµν around some metric f which is sometimes called
the absolute metric [Hint11]. This additional metric is often taken to be non-
dynamical and flat, but does not need to be. Theories in which this metric is
dynamical are called bi-gravity theories, and can be even further extended to
multi-gravity theories with multiple metrics. In the rest of this essay we shall
always take fµν to be non-dynamical.
Expressed in terms of these two metrics the simplest mass term we can add
is
Smass = dD
x −f
1
4
fµα
fνβ
m2
(hµνhαβ − hµαhνβ) , (27)
giving a total action
S = dD
x
√
−gR + −f
1
4
fµα
fνβ
m2
(hµνhαβ − hµαhνβ) . (28)
If we set fµν = ηµν, we obtain the linear FP mass term from the action (12),
and the total action becomes
S = dD
x
√
−gR −
1
2
m2
(hµνhµν
− h2
) . (29)
It is conventional to raise and lower indices with the absolute metric fµν rather
than the full gµν.
We can extend the mass term to include higher order terms by introducing an
arbitrary interaction potential U(f, h) which reduces to (27) at quadratic order,
giving an action of the form
S = dD
x
√
−gR + −f
1
4
fµα
fνβ
m2
U(f, h) . (30)
The potential U(f, h) can include all possible Lorentz-invariant combinations of
hµν to arbitrary degree. In particular, we can expand the interaction potential
as a series in h,
U(f, h) = U2(f, h) + U3(f, h) + U4(f, h) + U5(f, h) + . . . , (31)
4We would also need to add a similar source term, which we will leave out throughout this
discussion.
9
11. where Un(f, h) contains n-th order interactions of h. In general the n-th term
is a linear combination of all possible Lorentz contractions of n copies of hµν.
For example, the lowest order terms can be written as
U2(f, h) = [h2
] − [h]2
,
U3(f, h) = c1[h3
] + c2[h2
][h] + c3[h3
],
U4(f, h) = d1[h4
] + d2[h3
][h] + d3[h2
]2
+ d4[h2
][h]2
+ d5[h]4
, (32)
where the square brackets denote a trace taken with the metric f and ci and di
are numerical coefficients. The second order term is fixed by the requirement
that to second order the potential matches with (27). Since there is no obvious
constraint on the form of the interaction, the key will be to find coefficients for
which the massive gravity theory is consistent and free of the problems we are
about to discuss in the next section.
3 Main challenges of massive gravity
We now discuss the three main challenges of massive gravity. First we show how
the vDVZ discontinuity manifests in two different ways, by studying scattering
amplitudes and by studying the solution for a static point source. We then
expose the origin of the vDVZ discontinuity through the St¨uckelberg formalism,
a trick for restoring gauge symmetry by introducing additional fields. In section
3.2, we discuss the presence of an additional scalar degree of freedom, which for
the specific case we will study is a ghost known as the Boulware-Deser ghost. For
this we first employ the ADM formalism, in which the extra degree of freedom
shows up but the ghost character is not explicit. This becomes explicit when
we make use of the St¨uckelberg trick once again. The final section 3.3 of this
section discusses the effective field theory description of massive gravity. We
show how massive gravity differs from GR in this respect, and explain why this
is a problem for massive gravity.
3.1 vDVZ discontinuity
There are various ways in which the vDVZ discontinuity shows up. We first
present it as a discrepancy between graviton scattering amplitudes, following
[CarrGiu01], which is also the way in which it was originally presented by van
Dam and Veltman. To begin with, we need expressions for the propagators of
both the massive and massless theories. They do not agree with each other in
the massless limit of the massive propagator, as we shall see. The propagator is
the inverse of the differential operator for the quadratic part of the action. Let
us work in D = 4 throughout this discussion. We must write the FP action in
the form
S
(m=0)
F P = d4
x
1
2
hµνEµν,ρσ
(m=0)hρσ (33)
for some second order differential operator Eµν,ρσ
(m=0) and find its inverse on the
space of symmetric tensors. We claim that the differential operator is given by
Eµν
ρσ (m=0) = η(µ
ρην)
σ − ηµν
ηρσ ( −m2
)−2∂(µ
∂(ρη
ν)
σ)+∂ρ∂σηµν
+∂µ
∂ν
ηρσ. (34)
10
12. Indeed, acting on hµν
gives
1
2
Eµν
ρσ (m=0)hρσ
= h(µν)
− ηµν
h − m2
(h(µν)
− ηµν
h) − ∂(µ
∂ρhν)ρ
− ∂(µ
∂σhν)σ
+ ηµν
∂ρ∂σhρσ
+ ∂µ
∂ν
h
= (hµν
− ηµν
h) − ∂µ
∂λhλν
− ∂ν
∂λhλµ
+ ηµν
∂ρ∂σhρσ
+ ∂µ
∂ν
h − m2
(hµν
− ηµν
h), (35)
which matches with (14) upon contracting with hµν.
We state that the propagator is then given by5
D
(m=0)
ρσ,αβ =
1
p2 + m2
1
2
(PραPσβ + PρβPσα) −
1
3
PρσPαβ . (36)
where Pαβ ≡ ηαβ +
pαpβ
m2 should be understood as the induced metric on the
hyperboloid p2
= m2
. Since we are interested in on-shell processes, the induced
metric will be used to raise and lower indices in massive theory scattering pro-
cesses.
Now consider the massless theory and fix the gauge to be (10). The massless
FP action in the absence of matter then takes the simple form
S
(m=0)
F P = d4
x
1
2
hµν hµν
−
1
4
h h , (37)
which can be written in the form S
(m=0)
F P = d4
x1
2 hµνEµν,ρσ
(m=0)hρσ with the dif-
ferential operator
Eµν,ρσ
(m=0) =
1
2
(ηµρ
ηνσ
+ ηµσ
ηνρ
) −
1
2
ηµν
ηρσ
. (38)
The inverse of this operator is easier to derive and gives the propagator6
D
(m=0)
ρσ,αβ =
1
p2
1
2
(ηραησβ + ηρβησα) −
1
2
ηρσηαβ . (39)
Notice that not only has the metric been replaced by the flat metric, but also
the coefficient of the last term here is 1
3 rather than 1
2 . This difference is re-
sponsible for the discrepancy we now show.
Consider two sources Tµν
and T µν
interacting via the exchange of a graviton.
The interaction amplitude for such a process for a generic propagator Dµν,ρσ is
given by ([Zee10],[CarrGiu01])
A = G Tµν
Dµν,ρσT ρσ
, (40)
where G is the gravitational constant. Upon substituting (36) we find for the
massive theory
A(m=0)
=
G
p2 + m2
Tµν
T µν
−
1
3
Tµ
µ T µ
µ , (41)
5Cf. section 2.3 in [Hint11]. Showing that it is the inverse of the differential operator
constitutes a long calculation which is not given in the reference.
6Note that the inverse we are looking for is the inverse on the space of symmetric tensors.
The identity takes the form 1
2
(ηµ
αην
β + ην
βηµ
α).
11
13. while in the massless theory substituting (39) gives
A(m=0)
=
G
p2
Tµν
T µν
−
1
2
T µ
µ T ν
ν . (42)
Hence taking the limit m → 0 in A(m=0)
gives a scattering amplitude which still
differs from A(m=0)
in the coefficient of the last term. Scattering amplitudes
are physical predictions which can be tested experimentally. A disagreement
between the two scattering amplitudes means that only one of the theories can
be correct.
It is interesting to note that this discontinuity is inherent to the spin 2 theory
of massive gravity. A similar analysis can be performed for the spin 1 the-
ory, and it is found that there is no mismatch between the amplitudes of the
massless theory and the amplitude of the massive theory in the massless limit
(cf. section 2.1.2. in [dR14]). In other words, there exist massive photon the-
ories whose predictions approach classical electrodynamics in the massless limit.
The second approach to understanding the vDVZ discontinuity is by study-
ing the Newtonian potential and light bending angle predicted by both theories.
These quantities can be derived from studying the gravitational field produced
by a point particle of mass M at rest in the origin. We can solve the field equa-
tions (15) explicitly for this source (cf. section 3.2 in [Hint11]). The solutions
for hµν can then be related to the Newtonian potential φ and light bending
angle α through the PPN formalism7
and we find for the massive theory
φ = −
GM M
r
, α =
4GM M
b
, (43)
where GM plays the role of the gravitational constant for the massive theory
and may be different from Newton’s gravitational constant. On the other hand,
the massless theory gives
φ = −
4
3
GM
r
, α =
4GM
b
, (44)
where G is Newton’s constant of gravitation. We see that it is not possible
to choose GM such that both the Newtonian potential and the angle of bend-
ing agree with GR. For example, we can set GM = 4
3 G to have agreement of
the potentials, but then the bending angle would be off by a factor 4
3 . Since
experimental predictions of GR have been verified to high precision, we might
conclude at this point that massive gravity cannot be the correct theory. How-
ever, since we have only looked at the linear theory, there is hope that the vDVZ
discontinuity disappears in the non-linear theory. Before going to the non-linear
theory, let us first try to understand the origin of the discontinuity.
Origin of the vDVZ discontinuity
We can better understand the nature of the discontinuity by employing the
St¨uckelberg formalism. This is a method of introducing additional fields into
7PPN stands for Parametrized Post-Newtonian. The bending of light is expressed in terms
of a PPN parameter.
12
14. the theory, so that it becomes explicit how the degrees of freedom for the mas-
sive graviton arrange themselves as a tensor, a vector and a scalar. Under a
suitable limit these will decouple from each other, but the scalar will not be
completely free, as we shall see.
Let us first go back to the linear FP action
S = dD
x L(m=0) −
1
2
m2
(hµνhµν
− h2
) + κhµνTµν
, (45)
where we have isolated the part of the Lagrangian describing a free massless
graviton into the term L(m=0). We now introduce an additional field Aµ by
making a field redefinition of hµν patterned after its gauge symmetry in the
m = 0 theory:
hµν → hµν + ∂µAν + ∂νAµ (46)
The action then becomes
S = dD
x L(m=0) −
1
2
m2
(hµνhµν
− h2
) − 2m2
(hµν∂µ
Aν
− h∂µ
Aµ)
+
1
2
m2
FµνFµν
+ κhµνTµν
− 2κAµ∂νTµν
. (47)
This action has the gauge symmetry
hµν → hµν + ∂µξν + ∂νξµ, Aµ → Aµ − ξµ, (48)
where ξµ is a gauge parameter. Indeed, L(m=0) is invariant under this gauge
transformation, while the remaining terms transform as
δ(−
1
2
hµνhµν
) = −δhµνhµν
−
1
2
δhµνδhµν
= −2∂µξνhµν
+ ξν ξν
− (∂ν
ξν)2
δ(
1
2
h2
) = h δh +
1
2
(δh)2
= 2∂µξµ
h + 2(∂µξµ
)2
δ(−2hµν∂µ
Aν
) = −2δhµν∂µ
Aν
− 2hµν∂µ
δAν
= 2ξν Aµ
− 2(∂µ
ξµ)(∂ν
Aν) + 2hµν∂µ
ξν
δ(2h∂µAµ
) = 2δh∂µAµ
+ 2h∂µδAµ
= 4(∂µξµ
)(∂νξν
) − 2h(∂µ
ξµ)
δ(
1
2
FµνFµν
) = δFµνFµν
+
1
2
δFµνδFµν
= −ξν ξν
− (∂ν
ξν)2
− 2ξν Aν
− 2(∂µAµ
)(∂νξν
). (49)
Here we have regarded terms which are related by partial integration as equiv-
alent. Adding up the terms in (49) gives 0. We have thus restored the gauge
symmetry of the massless action by introducing an additional field. We can
normalize the kinetic term of this field by setting Aµ → 1√
2m
Aµ. If we assume
the energy-momentum tensor is conserved and take the m → 0 limit, we find
S = dD
x L(m=0) −
1
4
FµνFµν
+ κhµνTµν
. (50)
13
15. Hence Aµ describes a massless field, and can be interpreted as a set of Goldstone
fields. We also see that Aµ decouples from hµν and represents a free massless
spin 1 particle which has two degrees of freedom. Therefore, we now have an
action which describes four degrees of freedom, one short of the five degrees
of freedom for the massive spin 2 particle. The solution is to introduce an
additional scalar field φ by the following transformation modeled after the gauge
symmetry of Aµ:
Aµ → Aµ + ∂µφ. (51)
This gives the action
S = dD
x L(m=0) −
1
2
m2
(hµνhµν
− h2
) − 2m2
(hµν∂µ
Aν
− h∂µ
Aµ)
−2m2
(hµν∂µ
∂ν
φ +
1
2
m2
FµνFµν
− h φ)
+κhµνTµν
− 2κAµ∂νTµν
+ 2κ∂µφ∂νTµν
} . (52)
Note that this action has the additional gauge symmetry
Aµ → Aµ + ∂µΛ, φ → φ + Λ, (53)
where Λ is an arbitrary scalar function. This is easily checked by considering
the gauge transformations of the separate terms:
−δ(hµν∂µ
Aν
− h ∂µAµ
) = −hµν∂µ
∂ν
Λ + h Λ
−δ(hµν∂µ
∂ν
φ − h ∂µ∂µ
φ) = hµν∂µ
∂ν
Λ − h Λ
−δ(Aµ ∂νTµν
) = −∂µΛ ∂νTµν
δ(∂µφ ∂νTµν
) = ∂µΛ ∂νTµν
.
We can fix the gauge to be φ = 0, Aµ = 0 through the gauge symmetries (48) and
(53). This is called the unitary gauge and reproduces the original action (45).
Hence the entire processes of adding new fields has only resulted in additional
redundancy in our description, and the new action is equivalent to the original
action.
After suitable rescaling (in addition to Aµ → 1√
2m
Aµ we also take φ → 1
m2 φ),
we take the m → 0 limit and find that the action takes the form
S = dD
x L(m=0) −
1
4
FµνFµν
+ −2(hµν∂µ
∂ν
φ − h∂2
φ) + κhµνTµν
. (54)
We can now decouple the fields of the action (54) by performing the field trans-
formation
hµν → hµν +
2
D − 2
φηµν, (55)
After normalizing φ, we eventually find the action
S = dD
x L(m=0) −
1
4
FµνFµν
+ −
1
2
∂µφ∂µ
φ + κhµνTµν
+
1
(D − 1)(D − 2)
κφTµ
µ . (56)
14
16. Note that we have decoupled our fields by taking a massless limit m → 0 and
performing a field redefinition. In the non-linear theory the massless limit will
not suffice to decouple the fields as we shall see. From the action we see that
the action has separate terms for a scalar field φ, a massless vector field Aµ
and a tensor field hµν. These fields do not couple to each other, but the scalar
couples to the trace of the energy-momentum tensor. This is the origin of the
vDVZ discontinuity. The scalar is not simply a free scalar field, but its solution
is dependent on the presence of matter. Note that φ affects the physical metric,
through the transformation (55): here hµν before the transformation is the
metric perturbation in massive gravity, while hµν after the transformation is
described by the FP action and hence is the metric perturbation in GR. The
additional contribution from φ means that our theories deviates from GR.
3.2 Boulware-Deser ghost
We now return to the NLMG action with flat absolute metric fµν = ηµν, given
by (29). It appears that this action contains an additional degree of freedom.
This was discovered by Boulware and Deser in 1972 and the extra degree of
freedom manifests as a ghost, a scalar with a wrong sign for the kinetic term.
This ghost has become known as the Boulware-Deser (BD) ghost, and we shall
see how its presence can be inferred in two ways. The original way in which
Boulware and Deser discovered the ghost was by studying the theory in the
ADM formalism, which we shall discuss first. The ghost also appears in the
St¨uckelberg formalism discussed in the previous section as an additional mode
for the scalar.
Counting degrees of freedom can be done conveniently in the Hamiltonian
formalism. For massive gravity, we only need to treat the spatial components
of the metric hij and its associated momenta πij
≡ ∂L
∂hij
as dynamical. In total
they have 12 independent components. In the massless theory it can be directly
shown that h00 and h0i are non-dynamical (cf. section 2.1 in [Hint11]) and
therefore can be interpreted as Lagrange multipliers enforcing four constraints
(primary constraints). They generate an additional four gauge invariances (sec-
ondary constraints), leaving 4 phase space degrees of freedom corresponding to
the two real degrees of freedom of the photon. If the mass is not zero, the
components h0i are no longer non-dynamical, but can be solved in terms of the
πij
and substituted back into the action. However, h00 is still a Lagrange mul-
tiplier enforcing one constraint and generating another, leaving 10 phase space
degrees of freedom corresponding to the five polarization degrees of freedom of
the massive graviton.
For the NLMG theory we have to work in the ADM formalism. The start-
ing point is the 3+1 decomposition of GR for which we choose a foliation of
spacetime into spacelike slices Σt. This allows one to define the Lagrangian and
Hamiltonian as integrals over Σt. The ADM variables are the lapse function
N ≡ 1√
−g00
, shift functions Ni ≡ g0i and the spatial metric which we write as
γij = δij + hij. We shall write Nµ = (N, Ni) for convenience. Expressed in
15
17. terms of the ADM variables the metric components are
g00 = −N2
+ gij
NiNj, (57)
g0i = Ni, (58)
gij = γij. (59)
The variables γij and their canonical momenta πij ≡ δL
δ ˙γij
are the twelve phase
space variables, while the Nµ are auxiliary variables. A pair of conjugate vari-
ables can be interpreted as one propagating mode. Hence we start with six
propagating modes, and depending on the equations for Nµ there could be con-
straints on the phase space variables which reduce the number of propagating
modes.
The Einstein-Hilbert action in terms the Hamiltonian formalism becomes
SEH = dt
Σt
dd
x πij
˙γij − H , (60)
where H is the Hamiltonian. One can show that in ADM variables, the Hamil-
tonian is (cf. 6.4 in [Hint11])
H =
Σt
dd
x NC + NiCi
, (61)
where C and Ci are functions depending only on πij and hij ≡ γij − δij. Substi-
tuting this into (60) gives
SEH = dD
x πij
˙γij − NC + NiCi
. (62)
Since the functions C and Ci are linear in N and Ni, we see that Nµ takes the
role of a set of four Lagrange multipliers. Their equations of motion give four
constraints on the phase space variables. Another four degrees of freedom are
removed by the diffeomorphism invariance of GR, so four independent phase
space variables remain. These correspond to the two propagation modes of the
massless graviton.
On the other hand, for NLMG the action (29) in terms of ADM variables
takes the form
S = dD
x πij
˙γij − NC − NiCi
−
1
4
m2
hijhij + hiihjj + 2hii − 2N2
hii + 2NiNi − 2Ni
Ni , (63)
where C and Ci are the same functions as above. However, they now no longer
serve as constraints, because of the presence of quadratic terms in Nµ. Instead,
their equations of motion can be are linear in Nµ. This means that we can
directly solve them in terms of γij and πij
and substitute the solution back into
the action. This would leave an action in which the phase space variables are
not subject to any constraints. Therefore, the 12 phase space degrees of freedom
16
18. are all active and there are 6 polarization states.
The presence of the ghost becomes clearer in the St¨uckelberg formalism. We
introduce additional fields Y µ
(x) modeled after the diffeomorphism invariance
of the Einstein-Hilbert action (5) by making the replacement
gµν(x) →
∂Y α
∂xµ
∂Y β
∂xν
gαβ(Y (x)) ≡ Gµν(x). (64)
Upon replacing gµν by Gµν, and therefore also replacing hµν by Hµν ≡ Gµν −fµν
in the massive NLMG action (28), the resulting action becomes
S = M2
P dD
x
√
−gR + −f
1
4
m2
(Hµν
Hµν − H2
) . (65)
This action is invariant under the transformation
gµν(x) →
∂fα
∂xµ
∂fβ
∂xν
gαβ(f(x)), Y µ
(x) → f−1
(Y (x))µ
. (66)
This follows from the fact that the EH part is invariant under coordinate trans-
formations and Gµν is invariant under this transformation (and therefore also
Hµν is invariant). To verify the last claim, begin by performing the metric
transformation and then the transformation for Y µ
:
Gµν = ∂µY α
∂νY β
gαβ(Y (x))
→ ∂µY α
∂νY β
∂αfλ
(Y ) ∂βfσ
(Y )gλσ(f(Y (x)))
→ ∂µ f−1
(Y )
α
∂ν f−1
(Y )
β
∂αfλ
(f−1
(Y ))∂βfσ
(f−1
(Y ))gλσ(Y (x))
(67)
However, by the chain rule we have
∂µ f−1
(Y )
α
= ∂λ(f−1
)α
|Y ∂µY λ
, (68)
and by taking two terms from (67) together we get also by the chain rule
∂αfλ
|f−1(Y )∂µ f−1
(Y )
α
= ∂αfλ
|f−1(Y )∂λ(f−1
)α
|Y ∂µY λ
= ∂µ(fλ
|f−1(Y )) = ∂µY λ
. (69)
A similar expression holds for the other two terms and plugging the result back
into (67) gives
∂µ f−1
(Y )
α
∂ν f−1
(Y )
β
∂αfλ
(f−1
(Y ))∂βfσ
(f−1
(Y ))gλσ(Y (x))
= ∂µY λ
∂νY σ
gλσ(Y (x)) = Gµν(x).
We have thus restored diffeomorphism invariance in our theory. We then intro-
duce the field Aµ
by expanding Y µ
(x) = xµ
+ Aµ
(x). Now our expression for
Gµν becomes
Gµν = ∂µY α
∂νY β
gαβ = δα
µ + ∂µAα
δβ
ν + ∂νAβ
(fαβ + hαβ)
= gµν + ∂µAν + ∂νAµ + ∂µAα
∂νAα + . . . (70)
17
19. where we have put terms containing hµν into the ellipsis. Note that we raise
and lower indices with fµν. We subsequently introduce the scalar by replacing
Aµ → Aµ + ∂µφ. We insert this into (70) and using Hµν ≡ Gµν − fµν we
eventually obtain the following expression for Hµν:
Hµν = hµν −∂µAν −∂νAµ −∂µAα
∂νAα −2∂µ∂νφ−∂µ∂α
φ∂ν∂αφ−. . . . (71)
At the linear level this reduces to the replacement (46) for the linear theory we
studied before.
The full action becomes a series expansion in powers of φ, Aµ and hµν. In
analogy to the rescaling in the linear theory, we canonically normalize our fields
by introducing
ˆh = MP h, ˆA = mMP A, ˜φ = m2
MP φ, (72)
and perform a field redefinition
ˆhµν → ˆhµν + ηµν
˜φ (73)
The scalar, vector and tensor fields can now be decoupled, so that each is de-
scribed by a separate Lagrangian and there are no interaction terms. Recall that
in the linear case (section 3.1), we could decouple the fields by simply taking
the massless limit and performing a field transformation. In the non-linear case,
the analogous trick is to take the decoupling limit. In [dR13], the decoupling
limit is explained as “a special scaling limit where all the fields in the original
theory are scaled with the highest possible power of the scale in such a way that
the decoupling limit is finite.” (cf. chapter 8). In particular, the decoupling
limit does not change the number of degrees of freedom of the theory. It may
decouple them, so that the modes do not interact with each other, but this is
not necessarily the case, as we shall see in section (4.2). For the specific NLMG
theory we are studying, there is a relevant scale Λ5 which we shall discuss in
the next section. The decoupling limit holds this scale fixed and is given by
m → 0, MP → ∞, T → ∞, Λ5,
T
MP
fixed, (74)
In this limit, ˆhµν and ˆAµ become free fields, while ˜φ is described by the La-
grangian (in D = 4)
Sφ =
1
2
d4
x
3
2
(∂ ˜φ)2
+
1
Λ5
5
1
2
( ˜φ)3
−
1
2
( ˜φ)(∂µ∂ν
˜φ)2
+
1
MP
˜φT , (75)
The equation of motion for φ is
3 ˜φ +
1
Λ5
5
3
2
˜φ
2
−
1
2
˜φ,µν
˜φ,µν
− ∂µ∂ν
˜φ ˜φ,µν
=
1
MP
T. (76)
Despite the appearance of as many as six derivatives in certain terms, this is in
fact a fourth order equation ([Hint11],[BabDef13]). Four initial conditions are
required to solve it, twice as many as the two initial conditions needed for a
second order equation. This means that the scalar has two degrees of freedom,
and it can be shown (by Ostrogradski’s theorem) that one of these degrees of
freedom is a ghost ([Hint11],[dR13]).
18
20. 3.3 Massive gravity and GR as effective field theories
Einstein’s theory of general relativity is a classical theory. Massive gravity as
we have studied so far is also a classical theory. We have not worried about
quantum effects yet, and as long as the length scale at which these become im-
portant is small enough, there is reason to hope the theory we have studied is
valid for a reasonably large regime. In order to determine the scale at which
the quantum effects become important, we need an effective field theory (EFT)
description of our theory. Quantum effects are then taken into account by cor-
rection terms in the Lagrangian suppressed by an energy cutoff scale Λ. Let rQ
be the length scale at which quantum effects become important. It is inversely
related to the energy scale Λ. Another length scale in the problem is the length
scale at which non-linearities become important. This is called the Vainshtein
radius, which we will denote as rV . In order for the Vainshtein mechanism to
work, it is essential that rQ < rV .
Let us return to the NLMG action (28) and make the replacement hµν →
Hµν as before. For convenience let us take fµν = ηµν and D = 4 throughout
this section. The action becomes
S = d4
x
√
−gR +
1
2
ηµα
ηνβ
m2
(HµνHαβ − HµαHνβ) (77)
We then expand Hµν in terms of the St¨uckelberg fields as in (71). This gives a
series expansion in powers of hµν, Aµ and φ. We shall not be dealing with the
explicit full action, which becomes an infinite series, but let us note that the
interaction consists of terms of the form
∼ m2
M2
P
ˆhnh
(∂ ˆA)nA
(∂2 ˜φ)n˜φ , (78)
where nh, nA, nφ are integers, and the hats and tilde refer to the canonically
normalized fields (72). This comes from the fact that in the expansion for Hµν
(71) each instance of Aµ always goes together with one derivative and each
instance of φ with two derivatives.
Also define Λλ ≡ (MP mλ−1
)
1
λ with λ ≡
3nφ+2nA+nh−4
nφ+nA+nh−2 . Now we can write the
above term more conveniently as
∼ Λ
4−nh−2nA−3nφ
λ
ˆhnh
(∂ ˆA)nA
(∂2 ˜φ)nφ
. (79)
In this way, we see that the action becomes an expansion in the interaction
terms with coefficients Λ
4−nh−2nA−3nφ
λ . Alternatively, one can view Λλ as a
scale determining the energy scale suppressing the interaction term. In the
limit m MP , the interaction term which is suppressed by the smallest energy
scale is the cubic term ∼ ∂ ˜φ3
, which is suppressed by an energy scale
Λ5 = (m4
MP )(1/5)
. (80)
Hence we will call the NLMG theory we are dealing with Λ5 massive gravity.
This scale Λ5 is easily interpreted as the cutoff scale for the EFT, at which quan-
tum effects become important and the EFT description breaks down. However,
it is perhaps more precise to call this the strong coupling scale (cf. section 10.5
in [dR14]). The cutoff scale is the scale at which a theory breaks down and new
19
21. physics arises, which cannot be described by the old theory. For example, for GR
the cutoff scale is the Planck scale MP . In contrast, at the strong coupling scale
only the perturbative description of the theory breaks down. As discussed in
section 8.4 of [Hint11], Λ5 is the scale at which the tree-level graviton scattering
amplitude becomes of order one. However, the fact that tree-level calculations
breaks down only implies that the perturbative description is no longer valid.
Regardless of whether the scale is the cutoff or the strong coupling scale, it gives
a scale beyond which our theory cannot be trusted to give accurate results.
We now relate Λ5 to the radius at which quantum effects become important.
This can be derived by studying quantum correction terms which generically
take the form (cf. section 8.4 in [Hint11])
∼
∂q
(∂2 ˜φ)p
Λ3p+q−4
5
(81)
where p, q are integers. One can then study the solution for a central source of
mass M, for which the radius at which these corrections become important is
rp,q ∼
M
MP
p−2
3p+q−4
1
Λ5
. (82)
The highest value for this radius is found to be
rQ ∼
M
MP
1
3
1
Λ5
. (83)
Now consider the scalar part of the action after taking the decoupling limit,
given by (75). If we ignore the cubic term, i.e. remove it from the action, then
the equation of motion for φ is
˜φ(x) =
1
6
T(x)
MP
. (84)
For a static source of mass M at the origin (so we may replace → 2
), the
solution is
˜φ(x) =
1
24π
M
MP
1
r
∼
M
MP
1
r
. (85)
When is such an approximation valid? The cubic term has a factor relative
to quadratic term of the order ∂4 ˜φ
Λ5
5
∼ M
MP
1
Λ5
5
1
r5 . When this relative factor
becomes of the order one, the cubic terms can no longer be ignored and non-
linearities start having effect. This occurs when the distance is smaller than a
given distance
rV ∼ (
M
MP Λ5
5
)
1
5 = (
GM
m4
)
1
5 . (86)
This radius is the Vainshtein radius for massive gravity. We have now found two
distance scales at which certain effect become important. In particular, for Λ5
massive gravity we have rQ < rV (cf. section 8.4 in [Hint11]), so quantum ef-
fects become important before the linear theory breaks down and non-linearities
20
22. enter the theory.
How do these length scales relate to each other in GR? First, we try to solve
the Einstein field equations in vacuum, Gµν = 0, for a spherically symmetric
metric of the form
gµν = −B(r)dt2
+ C(r)dr2
+ A(r)r2
dΩ2
. (87)
We can reparametrize the radial coordinate to set A(r) = C(r). If we insert this
ansatz into the field equations, we get non-linear differential equations, which
can be solved by expanding B(r) and C(r) around a flat spacetime. By choosing
suitable initial conditions (cf. section 6.1 in [Hint11]), we eventually find
B(r) = 1 −
rS
r
(1 −
1
2
rS
r
+ . . .) (88)
C(r) = 1 −
rS
r
(1 +
3
8
rS
r
+ . . .), (89)
where rS = 2GM ∼ M
M2
P
is the Schwarzschild radius. We find that the series are
expansions in rS
r . When this factor becomes of order 1, the higher order terms
in the expansion for B(r) and C(r) become important, and this implies that
non-linearities become important (cf. section 6.1 in [Hint11]). Hence in GR the
Vainshtein radius coincides with the Schwarzschild radius.
Furthermore, in GR, the Planck scale is the scale at which the classical theory
breaks down. The Planck mass expressed in terms of fundamental constants is
MP =
c
G
. (90)
There are many ways to see why new physics must enter when energies become
of the order MP (see [Adler10] for an interesting discussion). The easiest way is
perhaps by considering the relative strength of the gravitational force compared
to the other forces. For example, consider two objects of mass M and electric
charge e at a distance r apart. The gravitational force and electromagnetic force
become of equal order of magnitude when
GM2
r2
∼
e2
r
, ⇔ M2
∼
e2
G
= α
c
G
= αM2
P , ⇒ M ∼
√
αMP . (91)
Here α ≡ e2
c ∼ 1
137 is the fine-structure constant, and
√
α ≈ 1
12 . Hence near the
Planck mass the two forces become of similar order of magnitude. Since QED
does not take into account gravity, it can no longer give accurate predictions.
It is predicted that the four fundamental forces merge into one at the Planck
scale, giving rise to unexplored new physics.
In GR, the distance at which the classical theory breaks down is related to
the Planck mass by rQ = 1
MP
. This can be seen from the fact that the Comp-
ton wavelength for a particle with mass m is given by λ = h
mc . In natural units
= c = 1 this means that λ ∼ 1
m . At distances smaller than the Compton
wavelength a quantum theory of gravity is needed to describe the theory. Since
21
23. GR can only describe particles with mass up to MP , at distances smaller than
1
MP
it can no longer be accurate as a classical theory. This scale is smaller than
the Schwarzschild radius (cf. section 6.1 in [Hint11]), meaning that within the
Schwarzschild radius there is a considerable region in which the linear theory
breaks down and non-linearities have to taken into account, yet we can safely
ignore quantum effects. Such a region is absent for the Λ5 theory we considered
above. It turns out that the existence of such a region is crucial for a theory
if we want it to be consistent with GR at small length scales. The Vainstein
mechanism which we will discuss next relies on non-linearities to restore conti-
nuity with GR. However, if the theory becomes swamped with quantum effects
before these non-linearities enter, we cannot rely on the Vainstein mechanism.
4 Proposed solutions to the challenges
We are now in the right position to discuss two constructions which provide
solutions to the challenges explained in the previous section, at least for a range
of situations. The first is the Vainshtein mechanism, which solves the vDVZ dis-
continuity problem by hiding the extra degree of freedom and restores continuity
with GR. The second is the dRGT construction, which gives a ghost-free theory
and raises the cutoff scale. The Vainstein mechanism can also be employed for
the dRGT theories to ensure continuity with GR.
4.1 Restoring continuity through the Vainshtein mecha-
nism
The Vainshtein mechanism can be used for a range of modified gravity theories.
As we have seen, the linear theory suffers from the vDVZ discontinuity, so we
wish to get rid of the discontinuity by constructing a suitable NLMG. The Vain-
shtein mechanism is a trick for doing this by effectively hiding the extra degree
of freedom by adding a non-linear term to the action. As we have seen earlier,
massive gravity as described by NLMG theories has a radius rV at which the
linear theory breaks down. Vainshtein’s idea is below this radius, we might hope
to restore GR by taking into account non-linear terms. In this section, we will
first study a general class of theories for which the Vainshtein mechanism works
and then look specifically at massive gravity.
Consider modifying GR in the following way: add to the Einstein-Hilbert
action a specific term linear in an additional scalar field φ, a generic term in-
volving non-linear intersections of φ and a matter coupling term. The family
of such theories is called the k-mouflage family and has a generic action of the
form
Sk−m = M2
P d4
x
√
−g R +
1
MP
φR + m2
KNL(φ, ∂φ, ∂2
φ, . . .)
+ Sm[g]. (92)
Here KNL(φ, ∂φ, ∂2
φ, . . .) is a generic non-linear term, containing self-interactions
of φ. We now wish to expand the action by expanding the metric around flat
space as gµν = ηµν + hµν. We will expand as follows: expand the Einstein-
Hilbert term to quadratic order to get the FP action, expand the second term
22
24. to linear order in hµν, do not expand the non-linear term and expand the matter
coupling term to obtain the only possible coupling term ∼ hµνTµν
. Then we
normalize by replacing hµν → 1
MP
ˆhµν. To linear order, R = ∂µ
∂ν
hµν − h, and
after partially integrating twice, we find the action
Sk−m = d4
x −
1
2
ˆhµν
Eαβ
µν
ˆhαβ + ˆhµν
φ,µν − ˆh φ + M2
P m2
KNL +
1
MP
Tµν
ˆhµν
.
(93)
We can then do the transformation ˆhµν → ˜hµν − ηµνφ to decouple the tensor
from the scalar. The action becomes
Sk−m = d4
x −
1
2
˜hµν
Eαβ
µν
˜hαβ +
3
2
φ φ + M2
P m2
KNL +
1
MP
Tµν
˜hµν
− Tφ .
(94)
Now the tensor is described by precisely the same action as in linearized GR,
while the scalar action is non-linear and in addition couples to matter through
the last term. The equations of motion decouple into
Eαβ
µν
˜hαβ =
Tµν
MP
, (95)
3 φ + Eφ =
T
MP
, (96)
where Eφ ≡ M2
P m2 δKNL
δφ . Since KNL is a non-linear term containing terms
which are at least cubic in φ, Eφ is at least quadratic in φ. Note that the
equation for ˜hµν is identical to the equation of motion in linearized gravity and
therefore if ˆh ∼ ˜h then GR is restored.
Just as for massive gravity, there exists a radius rV = 1
Λn
(MP rS)
k−1
n for the k-
mouflage family at which the linear approximation is no longer valid. It depends
on the precise form of the interactions in Eφ through the values of k and n (cf.
section 5.1 in [BabDef13]).
Outside this radius, the linear theory is valid and Eφ φ, so the equations of
motion are well-approximated by
Eαβ
µν
˜hαβ =
Tµν
MP
, (97)
3 φ =
T
MP
. (98)
This tells us that φ is of the same order as ˜h. But they were related to the
physical metric through the transformation ˆhµν → ˜hµν − ηµνφ, so φ gives a
correction of the same order as the ˜hµν, the metric perturbation for GR. Hence
in the linear regime the theory deviates from GR.
Inside the Vainshtein radius, the non-linear term becomes dominant and the
equations of motion are approximated by
Eαβ
µν
˜hαβ =
Tµν
MP
, (99)
Eφ =
T
MP
. (100)
23
25. Since Eφ is at least quadratic in φ, the corrections are no longer of order ∼ ˜h
and can be neglected. Therefore, GR is restored inside the Vainshtein radius.
Let us now study massive gravity with a generic interaction term (see (30)).
It turns out that after taking the decoupling limit the action for the scalar takes
the form (cf. section 3.3 in [BabDef13])
S =
1
2
d4
x
3
2
˜φ ˜φ +
1
Λ5
5
α( ˜φ)3
+ β( ˜φ˜φ,µν
˜φµν
, ) +
1
MP
T ˜φ , (101)
where α and β are numeric coefficients. For instance, for the specific action (29)
we studied before, which has the BD ghost, α = 1
2 , β = −1
2 as can be seen from
(75).
Comparing with the k-mouflage action (93), we see that (101) is a k-mouflage
action with the non-linear kinetic term KNL in this case being a cubic term.
Hence there is good hope that the Vainstein mechanism works. However, we
must confirm that it is indeed possible to find globally defined solutions for the
equations of motions. This turns out not to be true in general. For instance, the
standard Λ5 theory example (75) we studied in fact does not have well-defined
solutions8
! However, if we replace the absolute metric by the full metric in the
interaction term (27), we get an action for which the Vainshtein mechanism
does work. In particular, the equation of motion for ˜φ can be solved in both
regimes, giving [BabDef13]
˜φ =
MP
3
rS
r 1 + O(( r
rV
)5
) r > rV
−2
√
2MP
9
rS
r
r
rV
5
2
r < rV
(102)
We should note that this solution is continuous at r = rV and also a well-defined
solution for the metric can be obtained (cf. section 5.1.2 in [BabDef13]. The
solution for r < rV shows that for r rV the scalar is negligible. Therefore
GR is restored within the Vainshtein radius. We have finally found a NLMG
which is continuous with GR.
4.2 Removing the ghost and raising the cutoff through the
dRGT construction
The Vainshtein mechanism solves one of the major problems, but we still need
to address the problem of the ghost and the low cutoff limit of the effective field
theory. The first question is whether it is possible to construct massive gravity
theories which are ghost free by choosing a particular form for the interaction
of the NLMG theory (30). This appears to be possible and such theories were
constructed by de Rham and Gabadadze. It was first shown that in decoupling
limit the theory was ghost-free to quartic order in non-linearities [dRG10]. Sub-
sequent work showed that in the ADM formalism, this particular construction
can be proven to be ghost free up to all orders [HS11]. After claims of the ghost
8The mass term agrees with equation (18) in [BabDef13], and in section 5.1.2 it is explained
that there is no solution for such a mass term. However, the closely related mass term (19),
which contains the full rather than the absolute metric, does have a solution.
24
26. reappearing at quartic or even cubic order in the St¨uckelberg formalism, it was
finally shown in [dRGT11] that also in the St¨uckelberg formalism the theory
remains ghost free. Apart from resolving the ghost problem, it also raises the
cutoff scale sufficiently such that unlike Λ5 massive gravity, the Vainshtein ra-
dius becomes greater than the radius at which quantum effects enter the theory.
This ensures the validity of the Vainshtein mechanism. We will henceforth refer
to this particular construction as the dRGT construction. In this section we
shall work explicitly in D = 4.
The dRGT construction relies on a particular form of the interactions given
by
S = M2
P d4
x
√
−g R + 2m2
(e2(K) + α3e3(K) + α4e4(K) , (103)
with
e2(K) =
1
2
[K]2
− [K2
] (104)
e3(K) =
1
6
[K]3
− 3[K][K2
] + 2[K3
] (105)
e4(K) =
1
24
[K]4
− 6[K]2
[K2
] + 3[K2
]2
+ 8[K][K3
] − 6[K4
] . (106)
Here K ≡ I − g−1f with physical metric g and absolute metric f and α3, α4
are numerical coefficients. Hence the dRGT construction gives a two-parameter
family of actions for a particular choice of the interaction which includes terms
up to quartic order. We can apply the same procedure as in NLMG of introduc-
ing Goldstone fields through the St¨uckelberg formalism, expand fµν = gµν −Hµν
in terms of these fields and normalize the fields. The result is that the leading
cubic interaction term ∼ ∂3 ˜φ
MP m4 responsible for the Λ5 cutoff drops out and the
new cutoff becomes
Λ3 = (m2
MP )
1
3 . (107)
The decoupling limit for this action is9
m → 0, MP → ∞, Λ3 fixed. (108)
Next we perform the field redefinition
ˆhµν → ˜hµν − ηµν
˜φ − ˜α
∂µ
˜φ∂ν
˜φ
Λ3
3
. (109)
This will remove most interaction terms between the spin 0 and spin 2 fields,
9Recall that we are only studying source-free NLMG. It would be interesting to consider
how these results differ when we add source terms.
25
27. and the action then takes the form (cf. section 4.3 in [BabDef13])
S = d4
x −
1
2
˜hµν
Eαβ
µν
˜hαβ +
3
2
˜φ ˜φ +
˜α
Λ3
3
˜φ,µ ˜φ,ν
X(1)
µν
−
1
Λ6
3
˜α2
2
+
˜β
3
˜φ,µ ˜φ,ν
X(2)
µν +
˜β
Λ6
3
hµν
−
˜α
Λ3
3
˜φ,µ ˜φ,ν
X(3)
µν
+
1
MP
Tµν
˜hµν
− T ˜φ −
˜α
Λ3
3
Tµν
˜φ,µ ˜φ,ν
(110)
In this expression ˜α and ˜β are related to α3 and α4 in the original action through
˜α = 1 + α3 and ˜β = α3 + α4. The X
(n)
µν are expressions with n scalar fields φ
and 2n derivatives. In particular, in this case we have
X(1)
µν =
1
2
αρσ
µ
β
ν ρσ
˜φ,αβ
X(2)
µν = −
1
2
αργ
µ
βσ
ν γ
˜φ,αβ
˜φ,ρσ
X(3)
µν =
1
6
αργ
µ
βσδ
ν
˜φ,αβ
˜φ,ρσ
˜φ,γδ, (111)
where is the totally anti-symmetric Levi-Civita tensor. Notice that the scalar
and tensor are coupled through the X
(3)
µν term in this case. It appears that there
is no transformation which can decouple the fields10
.
The scalar interactions of (110) are proportional to the Lagrangians
L2 = −
1
2
(∂φ)2
,
L3 = −
1
2
(∂φ)2
[Π],
L4 = −
1
2
(∂φ)2
([Π]2
− [Π2
]),
L5 = −
1
2
(∂φ)2
([Π]3
− 3[Π][Π2
] + 2[Π3
]), (112)
where Πµν = ∂µ∂νφ and the square brackets denote traces. These are know as
Galileon terms and it can be shown that they give rise to second order equa-
tions of motion (despite being higher order in derivatives). This means that the
action (110) does not contain an extra degree of freedom and is ghost free (cf.
chapter 9 in [Hint11]). Hence the dRGT construction gives a theory which in
the decoupling limit does not have a ghost.
The question is now whether the same is true away from the decoupling
limit. Can it be shown in general that the dRGT construction gets rid of the
BD ghost? This analysis can be done in the ADM formalism discussed before.
Recall that in general we would have six degrees of freedom described by the
spatial metric γij and their conjugate momenta πij
. In general the lapse N and
shift Ni do not give constraints on these 12 canonical variables. In order to
reduce the number of modes to 5, we remove 2 Hamiltonian degrees of freedom.
The key to this is to show that the equations of motion for N and Ni only
10This claim is made in section 5.1.3. of [BabDef13] but refers to another article.
26
28. depend on three combinations of N and Ni, and the fourth equation gives an
additional constraint on the dynamical variables. Furthermore, this constraint
must give rise to an additional constraint, such that in total two phase space
variables are eliminated. This work was performed in [HasRos12] and has lead
to the conclusion that the action (103) is ghost free to all orders.
The last step in showing that the dRGT construction gives a consistent the-
ory of massive gravity is to show that the Vainshtein mechanism works. Here
the technical details are complicated by the fact that for ˜β = 0 the action (110)
does not fall in the k-mouflage family and hence does not allow for straight-
forward application of the analysis discussed above. We must make additional
assumptions to decouple the equations of motion and solve for the metric. Such
assumption are made in sections 4.2 and 5.3 of [BabDef13], where it is argued
that several of the terms in the equations of motion can be neglected. The
result is for both ˜β = 0 and ˜β = 0 we can solve the equations of motion, at
least for a range of parameter values of ˜α. We find that for ˜β = 0, ˜φ decays as
1
r
r
rV
inside the Vainshtein radius, while for ˜β = 0, ˜φ ∼ r2
, independent of rV .
Hence in both cases, the scalar becomes ineffective at short distances, just as in
massive gravity. The Vainshtein mechanism works for the dRGT construction.
5 Unresolved issues
We have seen examples above of how the Vainstein mechanism and dRGT con-
struction work to resolve the main problems of massive gravity. However, there
are still unresolved issues, some of which are related to the dRGT construction
and Vainshtein mechanism. We give a short summary of these issues, following
recent studies ([FGLM13], [dR14]).
Firstly, these two methods might work in certain circumstances, but it is
not certain they can be applied to general situations without problems. For
instance, we have only considered static, spherically symmetric sources for the
Vainshtein mechanism, and the question is whether it will still work in time-
dependent and less symmetric situations. These are relevant for the study of
various physical phenomena including gravitational waves and collapsing stars.
Indeed, it is not sufficient that the Vainshtein mechanism and dRGT construc-
tion only work for the limited range of solutions we have discussed. A natural
question is whether it is able to obtain cosmological solutions. This means that
we should look for a Friedmann-Lemaitre-Robertson-Walker (FLRW) metric,
or closely related variant. Such cosmological solutions have been studied in
[FGLM13], but it was found that any open FLRW-type solution has a ghost.
Two alternative attempts to construct ghost free cosmological solutions were
discussed. The first is to introduce a level of anisotropy in the configuration of
the St¨uckelberg fields. In this way anisotropic FLRW solutions which are ghost
free in certain parameter ranges could be found. The second involves introduc-
ing an extra degree of freedom to the hidden sector. For the two possibilities
considered it was found that only one could potentially lead to stable solutions,
but such a solution was not yet found.
Secondly, although the dRGT construction can raise the strong coupling
27
29. scale sufficiently for the Vainshtein mechanism to work, it still relies on the
effective field theory description of the theory. A complete theory of massive
gravity would require knowledge about what happens beyond the cutoff. Also,
it has been argued that Λ3 is still relatively low [Hint11], and the hope is to
raise the cutoff to a scale comparable to the Planck scale. This would give more
insight in the cosmological constant naturalness problem. A UV completion of
the theory is also needed to understand the phenomenon of superluminal prop-
agation discussed below.
There seems to be evidence that superluminal propagation in theories closely
related to massive gravity. Galileons, which are described by terms such as (112)
have become a separate subject of study and it has been pointed out that in
certain cases superluminal propagation is predicted in Galileon theories. In
particular, it has been shown that they exhibit superluminal propagation for all
parameter values for which the Vainshtein mechanism works. Since the Galileon
solutions are closely related to massive gravity, it has been suggested that super-
luminal propagation also occurs in massive gravity. However, many examples
are refuted in [dR14] (cf. section 10.4), which points out problems such as lack
of global solutions and emphasizes studying the correct type of velocity to de-
termine whether causality is violated. The latter is an interesting remark. The
results of special relativity are only violated when the so-called front velocity,
the velocity at which the foremost piece of the disturbance moves, exceed the ve-
locity of light. This is because the front velocity determines the speed at which
information is transferred. On the other hand, the phase velocity and group ve-
locity are not related to the rate of information transfer and it has been known
for some time that there can exist situations where the group velocity is larger
than c. In practice, the front velocity is the k → ∞ limit of the group velocity
of waves of momentum k. Since the theories of massive gravity are classical and
no UV completion of the theories is known to date, the front velocity cannot be
determined from a classical theory. Thus the EFT description of the theory is
incapable of fully answering the question of whether superluminal propagation
occurs. The discussion in [dR14] shows that superluminal propagation is an
area of current research and debate, but a definite answer to whether it occurs
has yet to be found.
6 Conclusion
In this essay, we have discussed the theory of massive gravity and pointed out
three major problems involved in constructing such a theory: the vDVZ discon-
tinuity, the BD ghost and low-cutoff scales of the EFT. We have also discussed
how these can be resolved by the Vainshtein mechanism and dRGT construc-
tion. However, problems such as the limited range of applicable situations for
these constructions remain, and there are possibly also new problems such as
superluminal propagation. There is hope that some questions may be answered
by studying extra-dimensional theories, such as the DGP theory. Other related
approaches include studying Lorentz-violating theories and a class of theories
called New Massive Gravity (NMG). Turning to the question raised by the title
of the essay, the preliminary answer seems to be that at present it has not been
28
30. ruled out theoretically that the graviton could have a mass, but the theory of
massive gravity is still incomplete. We have not addressed the phenomenological
aspects of the theory in this essay, and eventually experimental evidence has to
provide evidence either supporting or ruling out the theory.
Acknowledgments
My thanks goes to Dr Yi Wang for discussing my essay plan and answering
questions about the essay. In addition, I have gained from the preliminary
essay meeting organized by Dr Wang and Dr Baumann.
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29
