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arXiv:hep-th/0505147v1 16 May 2005

HUTP-05/A0020

HD-THEP-05-09

UCB-PTH-05/14

LBNL-57558

Ghosts in Massive Gravity

Paolo Creminellia, Alberto Nicolisa, Michele Papuccib, and Enrico Trincherinic

aJefferson Physical Laboratory,

Harvard University, Cambridge, MA 02138, USA

bDepartment of Physics, University of California, Berkeley and Theoretical Physics Group, Ernest

Orlando Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

cInstitute for Theoretical Physics,

Heidelberg University, D-69120 Heidelberg, Germany

Abstract

In the context of Lorentz-invariant massive gravity we show that classical solutions around heavy sources

are plagued by ghost instabilities. The ghost shows up in the effective field theory at huge distances

from the source, much bigger than the Vainshtein radius. Its presence is independent of the choice of the

non-linear terms added to the Fierz-Pauli Lagrangian. At the Vainshtein radius the mass of the ghost

is of order of the inverse radius, so that the theory cannot be trusted inside this region, not even at the

classical level.

1 Introduction

In recent years there has been renewed interest in the possibility of giving a mass to the graviton. This

idea belongs to a broader class of proposals for modifying gravity at large distances.

theoretical interest, these models could be phenomenologically relevant as possible alternatives to dark

matter and dark energy. In this paper we reconsider the issue of the range of validity of massive gravity;

in particular we concentrate on the stability of classical solutions around massive sources.

The problem we want to address has a long history. Already in the first paper [1] Fierz and Pauli

observed that the mass term must be of the form m2

spectrum, besides the 5 degrees of freedom of the massive spin-2 graviton. A different structure would

result in an instability at an energy scale ∼ mg. Unfortunately Boulware and Deser showed that this

additional degree of freedom propagates when nonlinearities in the action are taken into account [2].

However, from an effective field theory point of view this is not necessarily a problem, until one specifies

the scale at which the ghost shows up, i.e. its mass. If this scale is above the UV cutoff Λ of the effective

theory this instability can be consistently disregarded.

Besides their

g(h2− hµνhµν), otherwise a ghost appears in the

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On the other hand, non-linearities of the classical theory are also the solution to the problem raised by

van Dam, Veltman, and Zakharov (vDVZ) [3, 4]: in the linearized theory predictions are not continuous

in the limit mg→ 0, because the helicity-0 component of the graviton does not decouple from matter.

However, Vainshtein [5] observed that the vDVZ discontinuity might not be relevant for macroscopic

sources because the linearized approximation around a source of mass M∗ breaks down at a distance

RV = (M∗M−2

full non-linear solution could be in perfect agreement with experiments. This is still an open issue in

massive gravity, but the Vainshtein effect has been shown to work in a closely related model, the DGP

model [6, 7, 8].

More recently massive gravity has been reconsidered in the effective field theory language, which

provides a systematic framework for dealing with quantum effects [9]. For this purpose it is useful to

restore the broken diffeomorphism invariance by introducing a set of Goldstone bosons. With this method

it is easy to see that the scalar longitudinal component of the graviton becomes strongly coupled at a

very low energy scale, much lower than what naively expected by analogy with the spin-1 case. In the

Fierz-Pauli theory the strong interaction scale is Λ5∼ (m4

Hubble parameter is (1011km)−1. By adding to the Fierz-Pauli Lagrangian a set of properly tuned

interactions of the form hn

In both cases the theory seems to lose predictivity at very large distances. For instance one can

wonder how this strong coupling affects the gravitational potential generated by an astrophysical source.

Apparently the potential is uncalculable at distances smaller than 1000 km; but in principle this could not

be the case. After all the strong coupling takes place in the Goldstone sector, and inside the Vainshtein

radius, if the Vainshtein effect applies, one expects the Goldstone to give a negligible correction to

the Newtonian potential. Whatever quantum effects take place at the cutoff distance, they could be

sufficiently screened from experiments. Nevertheless, without further assumptions, from an effective

theory point of view one should include in the Lagrangian all the possible operators allowed by the

symmetries and weighted by the cutoff. In this case the effective theory loses predictivity at a much

larger length scale: these higher dimension operators all become important at a huge distance from the

source when evaluated on the classical solution. In the improved theory with cutoff Λ3 this happens

at the corresponding Vainshtein radius RV ∼ (M∗M−2

means that we are unable to compute the gravitational potential at distances shorter than RV without a

UV completion. As a consequence there is no range of distances where nonlinear effects can be reliably

computed in the effective field theory. If we restrict to the original theory with cutoff Λ5the situation is

even worse. In this case the infinite tower of higher dimension operators become important at a distance

which is parametrically larger than the corresponding Vainshtein radius [9].

The picture looks very similar to the DGP model [11], where the same problems have been pointed out

[12]. In the DGP model our world is the 4D boundary of an infinite 5D spacetime. Gravity is described

by a standard 5D Einstein-Hilbert action with Planck mass M5 in the bulk and by an additional 4D

Einstein-Hilbert action localized on the boundary, with a much larger Planck mass M4. The resulting

Newton’s law is 4-dimensional below the critical length scale LDGP= M2

distances. From the 4D viewpoint there is a scalar degree of freedom, the brane bending mode π, whose

dynamics is closely related to that of the longitudinal Goldstone boson φ of massive gravity. In particular

strong interactions show up in the π sector at a tiny energy scale ΛDGP∼ (MP/L2

(taking LDGPof order of the present Hubble horizon H−1

possible operators allowed by the symmetries and suppressed by ΛDGP, around a heavy source they all

become important at the Vainshtein radius RV ∼ (M∗M−2

solution.

Pm−4

g)1/5which diverges for mg→ 0. Classical nonlinearities become important, and the

gMP)1/5, which for mgof order of the present

µνthe cutoff can be raised up to Λ3∼ (m2

gMP)1/3∼ (1000 km)−1[9, 10].

Pm−2

g)1/3. For the sun RV is ∼ 1016km. This

4/M3

5and 5-dimensional at larger

DGP)1/3∼ (1000km)−1

0). Also if one includes in the Lagrangian all

PL2

DGP)1/3when evaluated on the classical

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However all these difficulties depend on assumptions about the UV completion. In the DGP model

one can consistently assume a UV completion such that the effective theory is predictive down to distances

significantly shorter than 1/ΛDGP. For instance on the surface of the earth the cutoff can be pushed up

to ∼ cm−1, not far from the smallest length scale ∼ 100 µm at which gravity has been experimentally

tested. Of course a necessary requirement for this to be possible is the consistency of the classical theory:

in particular in DGP no classical instability develops in the π sector for all relevant astrophysical sources

and for a large class of cosmological solutions [13]. In this paper we want to study whether the same

stability properties hold for massive gravity. This is a basic consistency requirement one has to satisfy

before analyzing the theory at the quantum level and looking for a mechanism, analogous to that working

in DGP, that can make the theory predictive in a phenomenologically interesting range of scales.

The most convenient way to study the dynamics of the theory is to use the Goldstone formalism, that

we review in section 3; for our purposes the main interesting features of the model are encoded in the

Lagrangian of the longitudinal component φ of the Goldstone vector. If we start with a Fierz-Pauli mass

term, the dominant interactions for this scalar degree of freedom are cubic self-couplings with 6 derivatives

of the form (∂2φ)3. In the presence of a macroscopic source the Goldstone gets a non-trivial configuration

Φ(x); in order to study the stability of such a solution it is necessary to expand the action at quadratic

order in the fluctuations around Φ(x). It is evident that in general, because of the cubic self-coupling, the

fluctuations will get a higher-derivative kinetic term. As we discuss in section 2, this signals the presence

of a ghost-like instability already at the classical level. In DGP this does not happen: although the π

cubic self-coupling has 4 derivatives, its tensorial structure is such that fluctuations around a background

get only a 2-derivative kinetic term [12, 13]. Unfortunately, this does not work for the Fierz-Pauli theory.

Still, we have a large freedom in choosing the non-linear extension of FP, and one can wonder if it is

possible to cancel all higher derivatives terms and end up with a ghost-free theory. Sections 4 and 5

contain the answer: despite the freedom we have, ghost-like instabilities are unavoidable.

In the Goldstone language it is also easy to compute the scale at which this instability appears. Even in

the most favorable setup in which the cutoff is Λ3, the ghost enters in the effective field theory at distances

from the source parametrically larger than the (already huge) Vainshtein radius RV. Furthermore, when

in approaching the source we reach r = RV the mass of the ghost has dropped to 1/RV! This means

that in no way the theory can be extrapolated inside the Vainshtein radius.

The last part of the paper is devoted to discuss how the sickness of the theory is interpreted in the

unitary gauge. Clearly we expect the ghost we found to be the troublesome sixth degree of freedom.

With the Fierz-Pauli mass term, at quadratic level this degree of freedom does not appear because the

trace of the Einstein equations gives a constraint instead of a propagating equation. In section 6 we show

that this equation becomes dynamical in the presence of a curved background; we qualitatively estimate

in this simple case the mass of this new excitation and the result agrees with the mass we find for the

ghost in the Goldstone computation. Then, in section 7, we show in the Hamiltonian formalism that

there exists no non-linear extension of the Fierz-Pauli theory that can forbid the propagation of the sixth

mode in the presence of a slightly curved background. The analysis in the unitary gauge is powerful for

counting the number of degrees of freedom, but, unlike the Goldstone analysis, it says nothing about the

typical scales of these modes. Also in order to address stability issues one should study the positivity

of the Hamiltonian. This in general is difficult, and the analysis has been carried out by Boulware and

Deser for non-linear extensions of the form f(hµνhµν− h2) [2]. On the contrary the Goldstone analysis

concentrates from the very beginning on the strong interacting degree of freedom: all the interesting and

troublesome features of the theory are encoded in the dynamics of a single scalar field. This enormously

simplifies the analysis.

Recently it has been realized that massive gravity models with Lorentz violating mass terms can be

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significantly ‘healthier’ than the traditional Lorentz-invariant theory; it particular they can avoid the

vDVZ discontinuity and the strong coupling problem, and they can be free of ghosts [14, 15, 16]. In this

paper we stick to the Lorentz-invariant massive gravity theory.

2 Ghosts from higher derivative kinetic terms

Let us first be very specific about why higher derivative kinetic terms give rise to ghost-like instabilities.

Take for instance a massless scalar field φ with Lagrangian density (note that we are using the (−,+,+,+)

signature!)

L = −1

where Λ is some energy scale, a = ±1, and Vintis a self-interaction term. We show that, independently

of the sign of the second term, the system is plagued by ghosts. To do so we want to reduce to a purely

two-derivative kinetic Lagrangian, from which we know how to read the stability properties of the system.

We therefore introduce an auxiliary scalar field χ and a new Lagrangian

2(∂φ)2+

a

2Λ2(?φ)2− Vint(φ) , (1)

L′= −1

2(∂φ)2− a ∂µχ∂µφ −1

2a Λ2χ2− Vint(φ) , (2)

which reduces exactly to L once χ is integrated out. L′is diagonalized by the substitution φ = φ′− aχ.

We get

L′= −1

which clearly signals the presence of a ghost: χ has a wrong-sign kinetic term. Notice in passing that χ

can also be a tachyon, for a = −1: in this case χ has exponentially growing modes. But let us neglect

this possibility and concentrate on the ghost instability, which is unavoidable. A ghost, unlike a tachyon,

is not unstable by itself: its equation of motion is perfectly healthy at the linear level, and does not

admit any exponentially growing solution. The problem is that its Hamiltonian is negative, so that

when couplings to ordinary ‘healthy’ matter are taken into account (the potential term in our example

above) the system is unstable: with zero net energy one can indiscriminately excite both sectors, and this

exchange of energy happens spontaneously already at classical level. In a quantum system with ghosts in

the physical spectrum this translates into an instability of the vacuum. The decay rate is UV divergent

due to an infinite degeneracy of the final state phase space. It is not clear how to cutoff this divergence

in a Lorentz invariant way [17].

However the situation is not as bad as it seems: our ghost χ in eq. (3) has a (normal or tachyonic)

mass Λ, so that it will show up only at energies above Λ, i.e. when the four derivative kinetic term in

eq. (1) starts dominating over the usual two derivative one. We can consistently use our scalar field

theory eq. (1) at energies below Λ, and postulate that some new degree of freedom enters at Λ and takes

care of the ghost instability. For example, we can add a term −(∂χ)2to eq. (2) (for simplicity we stick

to the non-tachyonic case a = +1 and set Vint= 0),

2(∂φ′)2+1

2(∂χ)2−1

2a Λ2χ2− Vint(φ′,χ) ,(3)

LUV= −1

2(∂φ)2− ∂µχ∂µφ − (∂χ)2−1

2Λ2χ2.(4)

This drastically changes the high-energy picture, since the resulting Lagrangian obtained by demixing

now describes two perfectly healthy scalars, one massless and the other with mass Λ. At the same time,

at energies below Λ the heavy field χ can be integrated out from LUV, thus giving the starting Lagrangian

eq. (1) up to terms suppressed by additional powers of (∂/Λ)2. This example shows that in principle the

ghost instability can be cured by proper new physics at the scale Λ. In other words, eq. (1) makes perfect

sense as an effective field theory with UV cutoff Λ.

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3 The Goldstone action

In this section we briefly re-derive the Lagrangian of massive gravity along the lines of [9], i.e. keeping

explicit the Goldstone bosons of broken diffeomorphism invariance.

To write down a mass term for gravity, in addition to the full dynamical metric gµν, we have to take

a reference fixed metric: for our purposes we will take the Minkowski metric ηµν. A mass term breaks

invariance under general coordinate transformations. However, as it has been shown in Ref. [9], one

can always restore local coordinate invariance by the St¨ uckelberg trick: in analogy with massive gauge

theories one introduces a set of Goldstone fields and requires that they transform non-linearly under a

local coordinate transformation. The fundamental object to be used for this purpose is a symmetric

tensor Hαβ, built in terms of the reference metric, the field describing metric fluctuations hµν= gµν−ηµν

and the four Goldstone fields πµ,

Hαβ= hαβ+ ∂απβ+ ∂βπα+ ∂απγ∂βπγ. (5)

Hµνtransforms as a covariant tensor under local diffeomorphisms xα→ xα+ξαprovided that παshifts,

πα→ πα− ξα. As in non-abelian massive gauge theories, since now local coordinate invariance is non

linearly realized on the π field, a Lagrangian built using H will be valid as an effective theory and its

breakdown will appear as the Goldstone sector becoming strongly coupled at some scale Λ. This indeed

has been shown in Ref. [9]. It is useful to further split the 4 παfields into a vector and a scalar as

πµ= Aµ+ ∂µφ (6)

together with an additional hidden U(1) gauge invariance for Aµunder which φ shifts. Note that since φ is

a Goldstone boson under this U(1) gauge symmetry and πµis a Goldstone boson of broken diffeomorphism

invariance φ will appear with two derivatives in the Lagrangian, as evident from eq. (5). A mass term

for hµνcan be written down in terms of Hµνas

√−ggµνgαβ(aHµαHνβ+ bHµνHαβ) .(7)

Expanding H using (5) one easily realizes that for a generic choice of a,b there is a quadratic term

in φ containing 4 derivatives. This term signals the presence of a ghost as we have seen in Section 2.

Only for the Fierz-Pauli choice a = −b ≡ m2

does not have a kinetic term on its own, but only a kinetic mixing with hµν: m2

A conformal rescaling of the metric hµν=ˆhµν+ m2

gηµνφ diagonalizes this mixing and generates a small

(i.e. proportional to m2

the kinetic term is the origin of the low strong coupling scale as it enhances the φ interactions once the

fields are canonically normalized.

One can easily see that the most relevant interactions are of the form

gM2

Pthis four derivative term exactly cancels. In this case φ

gM2

P(∂µ∂νφhµν− ?φh).

g) kinetic term for φ besides interactions of the form φ(∂2φ)n. The smallness of

m2M2

P(∂2φ)3=(∂2φc)3

MPm4g

(8)

where φcis the canonically normalized field. These interactions saturate perturbation theory at the tiny

energy scale E ∼ Λ5≡ (m4

One can slightly improve the situation canceling these cubic interactions by adding H3terms. Now

the most relevant interactions will be of the form (∂2φ)4and this procedure can be repeated at any

gMP)1/5.

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