When is multimetric gravity ghost-free?

When is multimetric gravity ghost-free?

Kouichi Nomura and Jiro Soda

Department of Physics, Kyoto University, Kyoto 606-8502, Japan(Received 21 July 2012; published 26 October 2012)

We study ghosts in multimetric gravity by combining the minisuperspace and the Hamiltonian

constraint analysis. We first revisit bimetric gravity and explain why it is ghost-free. Then, we apply

our method to trimetric gravity and clarify when the model contains a ghost. More precisely, we prove

trimetric gravity generically contains a ghost. However, if we cut the interaction of a pair of metrics,

trimetric gravity becomes ghost-free. We further extend the Hamiltonian analysis to general multimetric

gravity and calculate the number of ghosts in various models. Thus, we find multimetric gravity with loop

type interactions never becomes ghost-free.

DOI: 10.1103/PhysRevD.86.084052 PACS numbers: 04.50.�h

I. INTRODUCTION

It is interesting to explore the possibility that a gravitonis massive from both a theoretical and phenomenologicalpoint of view. Theoretically, it is challenging because ofvarious difficulties in constructing a consistent model for amassive graviton. At the linear level, Fiertz and Paulisucceeded in constructing a ghost-free model for a massivegraviton [1]. However, it was soon recognized that thereis a tension between the theory and experiments, the so-called van Dam—Veltman-Zaharov discontinuity [2,3]. Itwas suggested that the nonlinearity resolves the vanDam—Veltman-Zaharov discontinuity [4]. Unfortunately,it turned out that the nonlinearity gives rise to a ghost, theso-called Boulware-Deser (BD) ghost [5]. Recently, deRham, Gabadadze, and Tolley have succeeded in construct-ing ghost-free nonlinear massive gravity theory [6,7] (seethe review [8] and references therein). Still, there remainsvarious theoretically intriguing issues to be explored.Phenomenologically, there is a chance to explain the cur-rent accelerating universe based on massive gravity. In fact,there appears to be an effective cosmological constantproportional to the square of graviton mass [9–14]. It isworth studying this possibility in detail.

One peculiar feature of a massive theory of the graviton isthe necessity of a reference metric which breaks the diffeo-morphism invariance. It is natural to promote this referencemetric to a dynamical variable, which is nothing but bimet-ric gravity. Bimetric gravity contains two metrics, g and f,interacting with each other. The history of bimetric gravityis long [15–17]. Curiously, bimetric gravity also suffersfrom the ghost problem. Thanks to the recent developmentin massive gravity, however, Hassan and Rosen have pro-posed ghost-free bimetric gravity [18–20]. A natural ques-tion arises whether or not we can construct ghost-freemultimetric gravity. Actually, a naive extension of bimetricgravity to the trimetric case was proposed in Ref. [21].There, three metrics, g, f, and h, have a pair interactionbetween ðg; hÞ, ðh; fÞ, and ðf; gÞ, which forms a loop struc-ture. In contrast to bimetric gravity, however, the presence

or the absence of the BD ghost remains unknown. Recently,Hinterbichler and Rosen showed that a large class of multi-vielbein gravity is ghost-free [22]. The relation to metrictheory is also discussed [23]. However, the relation to themodels presented in Ref. [21] is not clear. The differencebetween Ref. [21] and [22] comes from the loop typeinteraction. In fact, the proof by the vielbein method isnot applicable to the loop type interaction. Hence, weneed to study multimetric gravity with a different approach.In this paper, we propose a simple method to study the

ghost problem and clarify when multimetric gravity isghost-free. A method often used for the ghost analysis isto examine models in the decoupling limit. However, a morehonest way for probing ghosts is to use the Hamiltonianconstraint analysis using the Arnowitt, Deser, and Misner(ADM) formalism [24]. The difficulty in studying multi-metric gravity with the constraint analysis comes from theexistence of a shift vector. To avoid the difficulty, weemploy the minisuperspace approximation. The minisuper-space reduction of phase space makes the analysis simple.Nevertheless, it is sufficient to identify ghosts because thisreduction process does not fail to capture ghosts.The organization of the paper is as follows. In Sec. II, we

revisit bimetric gravity and explain our strategy for theghost analysis. In Sec. III, we investigate trimetric gravityusing our method and find that the loop type interactionallows ghosts. In Sec. IV, we further extend the analysis togeneral N -metric gravity. We clarify when ghosts appearin the spectrum. The final section V, is devoted to theconclusion.

II. BIMETRIC GRAVITY REVISITED

In this section, we revisit bigravity and explain ourmethod to probe a ghost. It is already known that bimetricgravity is ghost-free [19]. Here, we show the same conclu-sion can be obtained using a simpleminisuperspace approxi-mation. In the context of massive gravity, the decouplinglimit analysis turns out to be a useful method for the ghostanalysis. However, the most complete one is to use the

PHYSICAL REVIEW D 86, 084052 (2012)

1550-7998=2012=86(8)=084052(10) 084052-1 � 2012 American Physical Society

http://dx.doi.org/10.1103/PhysRevD.86.084052

Hamiltonian constraint analysis and count physical de-grees of freedom. Our strategy is to use the Hamiltonianconstraint analysis in the minisuperspace.

The action of ghost-free bimetric gravity [20] is given by

Sbi ¼ M2g

Zd4x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detgp

R½g� þM2f

Zd4x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detfp

R½f�

þ 2m2M2gf

Zd4x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detgp X4

n¼0

�nenðffiffiffiffiffiffiffiffiffiffiffig�1f

qÞ; (1)

where the first and the second terms are the Einstein-Hilbertaction for each metric g and f from which we can calculatethe scalar curvatures R½g�, R½f�. Here, we have two Planckmasses Mg and Mf. The last term describes the interaction

between two metrics and �n are dimensionless couplingconstants. The other constants m andMgf are introduced to

adjust the mass dimension. The square root of the matrix is

defined such thatffiffiffiffiffiffiffiffiffiffiffig�1f

p ffiffiffiffiffiffiffiffiffiffiffig�1f

p ¼ g��f��. The interactionterms are constructed by enðXÞ which we define, formatrix X,

e0ðXÞ¼1; e1ðXÞ¼ trX;

e2ðXÞ¼1

2ðtr2X� trX2Þ;

e3ðXÞ¼1

6ðtr3X�3trXtrX2þ2trX3Þ;

e4ðXÞ¼ 1

24ðtr4X�6tr2XtrX2þ3tr2X2þ8trXtrX3�6trX4Þ

¼detX; (2)

where we used the notation trnX¼ðtrXÞn and trXn¼trðXnÞ.It is useful to represent the interaction by the diagram inFig. 1. Note that there is the order between g and f which isdenoted by the arrow. It is known that the interactionproduces massless and massive gravitons and the spectrumis free of the Boulware-Deser ghost. This feature comesfrom a specific interaction form found in massive gravitytheory. Remarkably, there exists the diagonal diffeomor-phism invariance which indicates the presence of the mass-less graviton.

Now, let us perform the Hamiltonian constraint analysisbased on the ADM formalism. In particular, to make theanalysis tractable, we employ the minisuperspace approach.Namely, we assume spatial homogeneity and express met-rics in terms of ADM variables as

g��dx�dx� ¼ �NðtÞ2dt2 þ �ijðtÞdxidxj; (3)

where N is a lapse function and �ij is a spatial metric.

Similarly, we can take the following ansatz,

f��dx�dx� ¼ �LðtÞ2dt2 þ!ijðtÞdxidxj; (4)

where L is a lapse function and !ij is a spatial metric. It is

convenient to write them in a matrix form,

g�� ¼ �N2 0

0 �ij

!; g�� ¼ �1=N2 0

0 �ij

!;

f�� ¼ �L2 0

0 !ij

!; f�� ¼ �1=L2 0

0 !ij

!;

(5)

where �ij and!ij are inverse matrices of spatial metrics �ij

and !ij. Then, a basic part of interaction terms can be

calculated to be

ðg�1fÞ�� ¼ L2=N2 0

0 �il!lj

!;

ffiffiffiffiffiffiffiffiffiffiffig�1f

q¼ L=N 0

0ffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

p !

:

(6)

When we count physical degrees of freedom, the follow-ing must be taken into account. In the vacuum cases, wecan diagonalize one of two spatial metrics using diagonalspatial coordinate transformations. Performing a spatialcoordinate transformation xi ! �ðt0Þijxj, we can set one

spatial metric at the time t ¼ t0, �ijðt0Þ, a unit matrix �ij.

Moreover, since the orthogonal transformation does notchange �ijðt0Þ ¼ �ij, we can diagonalize _�ijðt0Þ simulta-

neously by using this freedom. At this stage, homogeneousspatial coordinates are completely fixed. Now, �ij and _�ij

are diagonal at the time t ¼ t0 as an initial condition. Thenwe assume the diagonal form of �ijðtÞ at all times, and

insert it into equations obtained from variations of action.Any contradiction never occurs in vacuum. Thus, we con-clude that one spatial metric �ijðtÞ can be diagonalized

because of the uniqueness of the solution. Hence, thenumber of components of one of two metrics reducesfrom 6 to 3. This fact will be used later.In this paper, for simplicity, we assume that interactions

are minimal [21,25], namely,

�0¼3; �1¼�1; �2¼0; �3¼0; �4¼1: (7)

Clearly, this simplification does not lose any generalityconcerning the ghost analysis. Then, the Lagrangian reads

L ¼ M2g�

ij _�ij þM2fp

ij _!ij � NCN � LCL; (8)

where �ij, pij are the canonical conjugate momentum of�ij, !ij. Here, we have defined

CN ¼ M2gffiffiffiffiffiffiffiffiffiffi

det�p

��ij�ij � 1

2�i

i�jj

��M2

g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffidet�ð3Þ

qR½��

þ a1ffiffiffiffiffiffiffiffiffiffidet�

p ðtrffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

q� 3Þ; (9)

FIG. 1 (color online). The interactionffiffiffiffiffiffiffiffiffiffiffig�1f

pcan be repre-

sented by a simple diagram. Each blob describes a spacetimewith a given metric. The arrow indicates the order of the product.

KOUICHI NOMURA AND JIRO SODA PHYSICAL REVIEW D 86, 084052 (2012)

084052-2

CL ¼ M2fffiffiffiffiffiffiffiffiffiffiffi

det!p

�pijpij � 1

2pi

ipjj

��M2

f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidet!ð3Þ

pR½!�

þ a1ðffiffiffiffiffiffiffiffiffiffidet�

p � ffiffiffiffiffiffiffiffiffiffiffidet!

p Þ; (10)

where the first two terms of constraints CN and CM come

from the Einstein-Hilbert term in the action, so ð3ÞR½�� andð3ÞR½!� are spatial scalar curvatures computed from � and!, respectively. The last term of each constraint comes fromthe interaction (see the derivation in Appendix A), and weuse a1 ¼ 2m2M2

gf. Now, the Hamiltonian is given by

H ¼ NCN þ LCL: (11)

Since there are two Lagrange multipliers, there are twoprimary constraints:

CN ¼ 0; CL ¼ 0: (12)

Moreover, we need to impose consistency conditions forthem,

_CN ¼ fCN;Hg ¼ LfCN; CLg � LCNL � 0;

_CL ¼ fCL;Hg ¼ NfCL; CNg � NCLN � 0; (13)

where the Poisson bracket fF;Gg is defined by

fF;Gg ¼�@F

@�mn

@G

@�mn �@F

@�mn

@G

@�mn

�

þ�

@F

@!mn

@G

@pmn �@F

@pmn

@G

@!mn

�: (14)

Here, ‘‘� 0’’ means ‘‘¼ 0’’ on the constraint surface.Notice that fF; Fg ¼ 0 because of spatial homogeneity.

To check if a secondary constraint arises or not, we haveto calculate the Poisson bracket CNL. From the calculationpresented in Appendix B, we obtain

CNL¼fCN;CLg¼a1

�1

2M2

g�ii�M2

f

ffiffiffiffiffiffiffiffiffiffiffidet�

det!

s �1

2pi

itrffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

q

� trðffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

qp!Þ

��: (15)

This leads to one secondary constraint CNL � 0. The con-sistency condition for the secondary constraint reads

_CNL ¼ NfCNL; CNg þ LfCNL; CLg � 0: (16)

This condition determines one of two Lagrange multipliersN and L. The remaining multiplier describes the diagonaltime reparametrization invariance in bimetric gravity.

The number of components of two metrics and theircanonical conjugates is 24. Since we can diagonalize oneof the two metrics, we should subtract 6 from this number.Recall that there are two primary constraints and onesecondary constraint. Furthermore, as we have one firstclass constraint, we have to put one gauge condition.Thus, the total number of degrees of freedom should beð24� 6� 2� 1� 1Þ=2 ¼ 7 in configuration space, whichmatches degrees of freedom of one massless graviton and

one massive graviton. This proves that the BD ghost isabsent in bimetric gravity described by the action (1).

III. TRIMETRIC GRAVITY

Now, we apply the method explained in the previoussection to trimetric gravity. In contrast to the bimetric gravity,there are two kinds of interactions, namely, the tree type andthe loop type interactions. We discuss both cases, separately.The action for trimetric gravity [21] can be written as

Stri ¼ M2g

Zd4x


R½g� þM2f

Zd4x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detfp

R½f�

þM2h

Zd4x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� dethp

R½h�

þ 2m21M

2gf

Zd4x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detgp X4

n¼0


qÞ

þ 2m22M

2fh

Zd4x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detfp X4

n¼0

�0nenð

ffiffiffiffiffiffiffiffiffiffiffif�1h

qÞ

þ 2m23M

2hg

Zd4x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� dethp X4

n¼0

�00nenð

ffiffiffiffiffiffiffiffiffiffiffih�1g

qÞ; (17)

where �n, �0n, and �00

n are free parameters and R½g�, R½f�,and R½h� are scalar curvatures constructed from metrics g,f, and h, respectively. We also introduced new mass pa-rameters m1, m2, m3, Mfh, Mhg and a Planck mass Mh. It

should be noted that there exists the diagonal diffeomor-phism invariance in this trimetric theory which makes oneof the gravitons massless. As discussed in Refs. [21,22],if this trimetric gravity contains no extra degrees of free-dom, the total number of degrees of freedom should be2þ 5þ 5 ¼ 12, which comes from one massless gravitonand two massive gravitons. From now on, we use

a1¼2m21M

2gf; a2¼2m2

2M2fh; a3¼2m2

3M2hg (18)

for notational simplicity. If we have a1 � 0, a2 � 0, anda3 � 0, all pairs ðg; fÞ, ðf; hÞ, and ðh; gÞ interact and wecall it the loop type interaction. When one of aiði ¼ 1; 2; 3Þis set to zero, two of three pairs of interactions remain,which we call the tree type interaction. The case where oneinteraction is cut is already proved to be ghost-free usingvielbein formalism [22]; however, for the loop type inter-action no one shows the presence or the absence of ghosts.In this paper, we settle this issue.Apparently, the full Hamiltonian constraint analysis is dif-

ficult. To circumvent this difficulty,we take themethodused inthe previous section. Namely, we assume spatial homogene-ity and express metrics in terms of ADM variables as

g��dx�dx� ¼ �NðtÞ2dt2 þ �ijðtÞdxidxj; (19)

where N is a lapse function and �ij is a spatial metric.

Similarly, we can take the following Ansatze,

f��dx�dx� ¼ �LðtÞ2dt2 þ!ijðtÞdxidxj; (20)

WHEN IS MULTIMETRIC GRAVITY GHOST-FREE? PHYSICAL REVIEW D 86, 084052 (2012)

084052-3

and

h��dx�dx� ¼ �QðtÞ2dt2 þ �ijðtÞdxidxj; (21)

where L andQ are lapse functions and!ij and �ij are spatial

metrics. To perform the Hamiltonian constraint analysis, weneed the Lagrangian in the ADM variables

L¼M2g�

ij _�ijþM2fp

ij _!ijþM2h

ij _�ij

�NCN�LCL�QCQ; (22)

where �ij, pij, and ij are the canonical conjugate momen-tum of �ij, !ij, and �ij. Here, three Hamiltonian constraints

CN ¼ M2gffiffiffiffiffiffiffiffiffiffi

det�p

��ij�ij � 1

2�i

i�jj

��M2

g


qR½��


p ðtrffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

q� 3Þ þ a3ð

ffiffiffiffiffiffiffiffiffiffidet�

p � ffiffiffiffiffiffiffiffiffiffidet�

p Þ;(23)

CL ¼ M2fffiffiffiffiffiffiffiffiffiffi

det!p

�pijpij � 1

2pi

ipjj

��M2

f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidet!ð3Þ

pR½!�

þ a2ffiffiffiffiffiffiffiffiffiffidet!

p ðtrffiffiffiffiffiffiffiffiffiffiffiffiffi!�1�

q� 3Þ þ a1ð


p � ffiffiffiffiffiffiffiffiffiffidet!

p Þ;(24)

and

CQ ¼ M2hffiffiffiffiffiffiffiffiffiffi

det�p

�ijij � 1

2i

ijj

��M2

h


qR½��


p ðtrffiffiffiffiffiffiffiffiffiffiffiffi��1�

q� 3Þ þ a2ð

ffiffiffiffiffiffiffiffiffiffidet!

p � ffiffiffiffiffiffiffiffiffiffidet�

p Þ(25)

emerge. The first line of each Hamiltonian constraint comes

from the Einstein-Hilbert term in the action, so ð3ÞR½��,ð3ÞR½!�, and ð3ÞR½�� are spatial scalar curvatures calculatedfrom �, !, and �, respectively. The other terms can bederived as explained in Appendix A. Then, the Hamiltoniancan be read off as

H ¼ NCN þ LCL þQCQ: (26)

Since there are three Lagrange multipliers, there arise threeprimary constraints:

CN ¼ 0; CL ¼ 0; CQ ¼ 0: (27)

Moreover, we need consistency conditions for them:

_CN ¼ fCN;Hg ¼ LfCN; CLg þQfCN; CQg � 0;

_CL ¼ fCL;Hg ¼ NfCL; CNg þQfCL; CQg � 0;

_CQ ¼ fCQ;Hg ¼ NfCQ;CNg þ LfCQ;CLg � 0:

(28)

To check if secondary constraints arise, we must calculatePoisson brackets. From the calculation in Appendix B, weobtain

CNL�fCN;CLg¼a1

�1

2M2

g�ii�M2

f


det!

s �1

2pi


q

� trðffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

qp!Þ

��: (29)

By performing permutations among g ¼ ðN;�Þ, f ¼ ðL;!Þ,and h ¼ ðQ;�Þ, we also get

CLQ�fCL;CQg¼a2

�1

2M2

fpii�M2

h

ffiffiffiffiffiffiffiffiffiffidet!

det�

s �1

2i

itrffiffiffiffiffiffiffiffiffiffiffiffiffi!�1�

q

� trðffiffiffiffiffiffiffiffiffiffiffiffiffi!�1�

q�Þ

��(30)

and

CQN �fCQ;CNg¼a3

�1

2M2

hii�M2

g


det�

s �1

2�i

itrffiffiffiffiffiffiffiffiffiffiffiffi��1�

q

� trðffiffiffiffiffiffiffiffiffiffiffiffi��1�

q��Þ

��: (31)

In general, quantities inside the bracket do not vanish. Hence,the coefficients a1, a2, and a3 determine the consistencyconditions.

A. Tree type interaction

In this subsection, we consider the tree type interaction

a1 � 0; a2 ¼ 0; a3 � 0; (32)

which cuts interaction between f and h as in Fig. 2. In anycase, there are primary constraints (27). Since CQL ¼CLQ ¼ 0 trivially holds, consistency conditions (28) lead

to equations

LCNLþQCNQ�0; NCLN �0; NCQN �0: (33)

Hence, we have two secondary constraints,

CNL � 0; CNQ � 0: (34)

Moreover, we must impose consistency conditions

FIG. 2 (color online). The diagram represents the tree typeinteraction.


084052-4

_CNL ¼ fCNL;Hg¼ NfCNL; CNg þ LfCNL; CLg þQfCNL; CQg � 0;

(35)

_CQL ¼ fCQL;Hg¼ NfCQL; CNg þ LfCQL; CLg þQfCQL; CQg � 0;

(36)

which determine two of three Lagrange multipliers N, L,and Q. The remaining multiplier is related to the gaugetransformation.

Eventually, we have five constraints and one gauge free-dom. In trimetric gravity, propagating modes are spatialmetrics. Each of them has six components, but as is alreadyexplained we can diagonalize one of them. Hence, trimetricgravity has 3þ 6þ 6 ¼ 15 degrees of freedom in configu-ration space and 15� 2 ¼ 30 in phase apace. Thus, thetotal number of degrees of freedom is ð30� 5� 1Þ=2 ¼ 12which matches the physical degrees of one massless andtwo massive gravitons. Therefore, no BD ghost exists in thespectrum. This conclusion is consistent with the one ob-tained by the vielbein method [22].

B. Loop type interaction

Now, we consider the more general loop type interactionrepresented by Fig. 3:

a1 � 0; a2 � 0; a3 � 0: (37)

It is obvious that

fCN;CLg�0; fCL;CQg�0; fCQ;CNg�0 (38)

even on the constraint surface. Hence, consistency condi-tions (28) do not generate any secondary constraint. Instead,it determines Lagrange multipliers N, L, and Q. However,due to the antisymmetric property of Poisson brackets

CNL ¼ �fCN; CLg ¼ �fCL; CNg ¼ CLN; (39)

CLQ ¼ fCL; CQg ¼ �CQL; (40)

CQN ¼ fCQ;CNg ¼ �CNQ; (41)

only two of them are determined. For example, choosing

L ¼ �CNQ

CNL

Q; N ¼ �CLQ

CLN

Q; (42)

all of consistency conditions (28) are satisfied.To conclude, we have three primary constraints and we

need one gauge condition to fix one undetermined Lagrangemultiplier which is associated with the time reparametriza-tion invariance. In trimetric gravity, as is already counted,there are 3þ 6þ 6 ¼ 15 degrees of freedom in configura-tion space and 15� 2 ¼ 30 in phase apace. In phase space,we have three constraints and one gauge condition, so thetotal number of degrees of freedom is ð30� 3� 1Þ=2 ¼13. If no BD ghost is present, there must be 2þ 5þ 5 ¼ 12degrees of freedomwhich come from one massless gravitonand two massive gravitons. Therefore, one extra degree offreedom exists and it should be a BD ghost. Thus, we haveproved the existence of a ghost in generic trimetric gravity.

IV. GENERAL MULTIMETRIC MODELS

Now, we are in a position to discuss more general cases.We explicitly calculate the number of ghosts if they exist.In this section, we consider N dynamical metrics gk

(k ¼ 1; 2; . . . ;N ) and interaction terms such as

XNk¼1

akffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detgk

p X4n¼0

�k;nenðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig�1k gkþ1

qÞ; (43)

where we define gNþ1 ¼ g1 and for later purpose we alsoneed g0 ¼ gN . Let us describe the interaction between twometrics gk and gkþ1 in terms of the ADM form of metrics

ds2k ¼ �N2kðtÞdt2 þ �k;ijðtÞdxidxj: (44)

Schematically, the interaction can be written as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detgkp X4

n¼0


qÞ

¼ NkFkð�k: �kþ1Þ þ Nkþ1Gkð�k: �kþ1Þ; (45)

whereFk andGk are some functions determined by parame-ters �k;n. Thus, the total interaction terms are given by

XNk¼1

akffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�detgk

p X4n¼0


qÞ

¼XNk¼1

NkfakFkð�k:�kþ1Þþak�1Gk�1ð�k�1:�kÞg: (46)

The Hamiltonian becomes

H ¼ XNk¼1

NkCk; (47)

Ck ¼ C0kð�k; �kÞ � akFkð�k: �kþ1Þ

� ak�1Gk�1ð�k�1: �kÞ; (48)FIG. 3 (color online). The diagram represents the loop typeinteraction.


084052-5

where C0k comes from the Einstein-Hilbert term for gk, so it

contains �k and its canonical conjugate momentum �k.Corresponding toN Lagrange multipliers, we haveN

primary constraints

Ck ¼ 0; ðk ¼ 1; 2; . . . ;N Þ: (49)

Next, we have to examine N consistency conditions

_C k ¼ Ck;k�1Nk�1 þ Ck;kþ1Nkþ1 � 0; (50)

where Ck;l ¼ fCk; Clg and Ck;l ¼ 0 if jk� lj ^ 2. In this

formula, N0 ¼ NN and NNþ1 ¼ N1 should be under-stood. Note that the explicit calculation gives rise to theimportant information

Ck;kþ1 / ak: (51)

The structure of this matrix depends on an odd or evennumber. For example, in the case N ¼ 4, we have

Ck;l ¼

0 C1;2 0 C1;4

�C1;2 0 C2;3 0

0 �C2;3 0 C3;4

�C1;4 0 �C3;4 0

0BBBBB@

1CCCCCA; (52)

while, in the case of N ¼ 5, we get

Ck;l ¼

0 C1;2 0 0 C1;5

�C1;2 0 C2;3 0 0

0 �C2;3 0 C3;4 0

0 0 �C3;4 0 C4;5

�C1;5 0 0 �C4;5 0

0BBBBBBBB@

1CCCCCCCCA: (53)

In the case of an odd number of metrics, we cannot split theequations into two independent sets, while, in the case ofan even number of metrics, we can split a set of equationsinto two independent groups of equations. Hence, we haveto discuss the two cases, separately.

A. Tree type interaction

First, we consider the tree type interaction.If we cut one of ðgk; gkþ1Þk¼1;2;...;N interactions as in

Fig. 4, for example, setting a1 ¼ 0, Eq. (50) leads toN �1secondary constraints

Ck;kþ1 � 0; ðk ¼ 2; 3; . . . ;N Þ; (54)

and their consistency conditions

_C k;kþ1 ¼XNl¼1

fCk;kþ1; ClgNl � 0; ðk ¼ 2; 3; . . . ;N Þ

(55)

determine N � 1 of Nk (k ¼ 1; 2; . . . ;N ), only oneLagrange multiplier remains undetermined. Therefore, thetotal number of degrees of freedom can be deduced as

1

2ð2ð3þ6ðN �1ÞÞ�N �ðN �1Þ�1Þ¼5ðN �1Þþ2;

(56)

which corresponds to N � 1 massive gravitons and onemassless. Therefore, there exists no BD ghost. This con-clusion is also consistent with the one obtained by thevielbein method [22].

B. Loop type interaction

Now, we come to our main point.If all of ðgk; gkþ1Þk¼1;2;...;N interactions exist as in Fig. 5,

the analysis gets a little complicated. We have to discussodd and even numbers, separately.

1. Odd number of metrics

First, we consider the case where N ¼ 2mþ 1, wherem is a natural number. In this case, we can classify Eq. (50)into the following four parts:

C2k;2k�1N2k�1 þC2k;2kþ1N2kþ1 ¼ 0 ðk¼ 1;2;3; . . . ;mÞ;(57)

C2k�1;2k�2N2k�2 þ C2k�1;2kN2k ¼ 0 ðk ¼ 2; 3; . . . ; mÞ;(58)

FIG. 4 (color online). The diagram represents the tree typeinteraction.

FIG. 5 (color online). The diagram represents the loop typeinteraction.


084052-6

C1;2mþ1N2mþ1 þ C1;2N2 ¼ 0; (59)

C2mþ1;2mN2m þ C2mþ1;1N1 ¼ 0: (60)

Solving Eq. (57), we see all of N2kþ1 (k ¼ 1; 2; . . . ; m) canbe expressed by N1. Similarly, Eq. (58) can be used toexpress N2k (k ¼ 2; 3; . . . ; m) in terms of N2. Substitutingthese results into Eqs. (59) and (60), we obtain a singleequation which determines N2 by N1. Thus, Eq. (50)determines N � 1 Lagrange multipliers, and one multi-plier is left undetermined, which reflects the existence ofgauge symmetry.

In the case of an odd number of metrics, there is nosecondary constraint, while we need one gauge conditionto fix the gauge degree of freedom. In conclusion, the totalnumber of degrees of freedom can be calculated as

1

2ð2ð3þ 6ðN � 1ÞÞ �N � 1Þ

¼ 5ðN � 1Þ þ 2þN � 1

2: (61)

Here, the first two terms correspond to massive and mass-less gravitons, respectively. The last one should be BDghosts and the number of ghosts is given by ðN � 1Þ=2.

2. Even number of metrics

Next, we consider the case N ¼ 2mþ 2, where m is anatural number. In this case, we can split Eq. (50) into twoindependent sets of equations,

Ck;k�1Nk�1 þCk;kþ1Nkþ1 ¼ 0 ðk¼ 1;3;5; . . . ;2mþ 1Þ;(62)

Ck;k�1Nk�1 þCk;kþ1Nkþ1 ¼ 0 ðk¼ 2;4;6; . . . ;2mþ 2Þ:(63)

The first set (62) contains only Nk ðk¼2;4;6;...;2mþ2Þ,and the second set (63) contains Nk ðk¼1;3;5;...;2mþ1Þ.Here, if the component Ck;k�1 is in Eq. (62), Ck�1;k ¼�Ck;k�1 must be in Eq. (63) and vice versa. Therefore, in

each set, every component Ck;k�1 appears only once. Now,

we define

Di;j ¼ C2i�1;2j; Mj ¼ N2j

ði; j ¼ 1; 2; 3; . . . ; mþ 1Þ: (64)

Note that Dij � 0 only for i� j ¼ 0, 1. Then, Eq. (62) can

be written asXj

Di;jMj ¼ 0; ði ¼ 1; 2; 3; . . . ; mþ 1Þ; (65)

which we can split into

D1;1M1 þD1;mþ1Mmþ1 ¼ 0; (66)

Di;i�1Mi�1þDi;iMi¼0; ði¼2;3; . .. ;mþ1Þ: (67)

Using Eq. (67), we can solve all ofMj ði ¼ 2; 3; . . . ; mþ 1Þin terms of M1. However, the relation between M1 andMmþ1 obtained from Eq. (67) is not the same as Eq. (66)because Eq. (67) contains no D1;1 and D1;mþ1. So, we have

to impose a constraint so that we get nontrivial Lagrangemultipliers. This is a secondary constraint expressed by

detDij ¼ 0: (68)

Under this condition, m of Mj ðj ¼ 1; 2; . . . ; mþ 1Þ are

determined, and one is left undetermined.Now, we take the latter set (63) and define

Ei;j ¼ C2i;2j�1; Wj ¼ N2j�1

ði; j ¼ 1; 2; 3; . . . ; mþ 1Þ: (69)

The same argument applies, soweget a secondary constraintdetEij ¼ 0, and one of Wj ðj ¼ 1; 2; . . . ; mþ 1Þ is left un-determined. However, matrix Eij satisfies Eij ¼ �Dji.

Hence, detEij ¼ 0 is not a new constraint. Therefore, from

Eq. (50), we get one secondary constraint detDij ¼ 0 and

two undetermined Lagrange multipliers. Then, we mustimpose a consistency condition for the secondary constraint

d

dtdetDij ¼

XNk¼1

fdetDij; CkgNk � 0; (70)

which reduces the number of undetermined Lagrangemulti-pliers from two to one.To summarize, there areN primary constraints and one

secondary constraint and we need one gauge condition.Thus, we come to the conclusion that the total number ofdegrees of freedom is

1

2ð2ð3þ 6ðN � 1ÞÞ �N � 1� 1Þ

¼ 5ðN � 1Þ þ 2þN � 2

2: (71)

Here, again, the first two terms correspond to massive andmassless gravitons, respectively. Hence, the number of BDghosts should be ðN � 2Þ=2.

C. More general diagrams

In the previous sections, we have considered tree and looptype interactions. In the case of bimetric gravity, the inter-action type is unique; namely, there is only the tree typeinteraction. In the case of trimetric gravity, there are twopossibilities, the tree and the loop type interaction. In thecase of tetrametric gravity, there are many loop type inter-actions represented by Fig. 6(a). If we cut some of theinteraction, we can make the tree type interaction and thebroom type interaction represented by Fig. 6(b). From ouranalysis, it is apparent that if the interaction contains at leasta loop, then there are ghosts. For example, the model in


084052-7

Fig. 6(c) contains a ghost. Therefore, in generic cases, thereexist ghosts in multimetric gravity. The number of ghostsdepends on the interaction pattern. To construct a viablemodel, we have to eliminate all of the loop type interactions.

V. CONCLUSION

We studied multimetric gravity by combining the minis-uperspace and the Hamiltonian constraint analysis. We firstrevisited bimetric gravity and explained why it is ghost-free.This proved the validity of our method. Then, we appliedour method to trimetric gravity and clarified when the modelcontains a ghost. We proved trimetric gravity genericallycontains a ghost. However, if we cut the interaction of a pairof metrics, trimetric gravity turned out to be ghost-free. Wefurther extended the Hamiltonian analysis to general multi-metric gravity and calculated the number of ghosts in vari-ous models. Thus, we found multimetric gravity with looptype interactions never becomes ghost-free. The number of

BD ghosts in the N -metric case turned out to beðN � 1Þ=2 or ðN � 2Þ=2, depending on whether the num-ber of metrics N is odd or even. Hence, the number of BDghosts increases by one every time two more metrics areintroduced. There are other models which may contain ghostsor may not contain any ghosts. It depends on the interactiontype. The number of ghosts can be calculated once thediagram characterizing the interaction pattern is given.Admittedly, what we have investigated is BD ghosts.

There may be other ghosts depending on the solutions[26,27]. In other words, the absence of BD ghosts is anecessary condition as a healthy model. In this paper, wehave studied interaction terms consisting of only pairs ofmetrics. However, as in Ref. [22], interactions of triplets orquadruplets may be allowed. We hope to study this possi-bility in the future. It is also interesting to extend ouranalysis to various models such as higher curvature theo-ries [28] or theories containing other fields [29].

ACKNOWLEDGMENTS

This work was supported in part by the Japan Society forthe Promotion of Science (JSPS) Grant No. 24-1693, theGrant-in-Aid for Scientific Research Fund of the Ministry ofEducation, Science and Culture of Japan No. 22540274, theGrant-in-Aid for Scientific Research (A) (No. 21244033,No. 22244030), the Grant-in-Aid for Scientific Research onInnovative Area No. 21111006, JSPS under the Japan-Russia Research Cooperative Program, and the Grant-in-Aid for the Global COE Program ‘‘The Next Generation ofPhysics, Spun from Universality and Emergence.’’

APPENDIX A: INTERACTION TERMS

In this appendix, we calculate the interaction terms separately. We use the following representations:

ðg�1fÞ�� ¼ L2=N2 0

0 �il!lj

!;


q¼ L=N 0

0ffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

p !

: (A1)

The first one is given by


e1ðffiffiffiffiffiffiffiffiffiffiffig�1f

qÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detg

ptr


q¼ ffiffiffiffiffiffiffiffiffiffi

det�p

NðL=N þ trffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

qÞ ¼ ffiffiffiffiffiffiffiffiffiffi

det�p ðLþ Ntr

ffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

qÞ: (A2)

Due to the combination g�1f, we got the linear terms with respect to the lapse functions. The second one becomes




p 1

2ðtr2


q� trðg�1fÞÞ ¼ ffiffiffiffiffiffiffiffiffiffi

det�p

N1

2fðL=N þ tr


qÞ2 � L2=N2 � trð��1!Þg

¼ ffiffiffiffiffiffiffiffiffiffidet�

p �Ltr


qþ 1

2Nðtr2


q� trð��1!ÞÞ

�: (A3)

Again, we obtained desired linearity for the lapse functions. The third one can be calculated as

FIG. 6 (color online). The diagram (a) represents the mostgeneral type interaction. The diagram (b) is the broom typeinteraction. The diagram (c) includes the loop type interaction,hence there should be a BD ghost.


084052-8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�detgp


qÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�detg

p 1

6ðtr3


q�3tr


qtrðg�1fÞþ2trðg�1fÞ3=2Þ


pN1

6fðL=Nþ tr


qÞ3�3ðL=Nþ tr


qÞðL2=N2þ trð��1!ÞÞþ2ðL3=N3þ trð��1!Þ3=2Þg


p �1

2Lðtr2


q� trð��1!ÞÞþ1

6Nðtr3


q�3tr


qtrð��1!Þþ2trð��1!Þ3=2Þ

�: (A4)

This is also linear with respect to the lapse functions. The last one is




pdet


q¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detfp ¼ L

ffiffiffiffiffiffiffiffiffiffiffidet!

p: (A5)

To sum up, the interaction terms readX4n¼0


qÞ ¼ N


p ��0 þ �1tr


qþ 1

2�2ðtr2


q� trð��1!ÞÞ þ 1

6�3ðtr3


q

� 3trffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

qtrð��1!Þ þ 2trð��1!Þ3=2Þ

�þ L

� ffiffiffiffiffiffiffiffiffiffidet�

p ��1 þ �2tr


q

þ 1

2�3ðtr2


q� trð��1!ÞÞ

�þ �4

ffiffiffiffiffiffiffiffiffiffiffidet!

p �: (A6)

Notice that all interaction terms are linear in L andN. This is the advantage of the minisuperspace model, which makes theHamiltonian constraint analysis simple.

In the paper, for simplicity, we always assume that interactions are minimal [21,25], namely,

�0 ¼ 3; �1 ¼ �1; �2 ¼ 0; �3 ¼ 0; �4 ¼ 1: (A7)

Clearly, this simplification does not lose any generality concerning the ghost analysis.

APPENDIX B: CONSTRAINTALGEBRA

In this appendix, we calculate a Poisson bracket. It is sufficient to look at the following:

fCN; CLg ¼�

M2gffiffiffiffiffiffiffiffiffiffi

det�p

�1

2�i

i�jj � �ij�ij

�;�a1


p �þ��a1


ptr


q;

M2fffiffiffiffiffiffiffiffiffiffiffi

det!p

�1

2pi

ipjj � pijpij

��

¼ a1

�M2

gffiffiffiffiffiffiffiffiffiffidet�

p� ffiffiffiffiffiffiffiffiffiffi

det�p

;1

2�i


��M2

f

ffiffiffiffiffiffiffiffiffiffiffidet�

det!

s �tr


q;1

2pi

ipjj � pijpij

��: (B1)

The point is that the result is proportional to a1. Each term can be manipulated as� ffiffiffiffiffiffiffiffiffiffidet�

p;1

2�i


�¼ @


p@�mn

@

@�mn

�1

2�i


�¼ 1

2


p�mnð�mn�

ii � 2�mnÞ ¼ 1

2


p�i

i (B2)

and

�tr


q;1

2pi

ipjj � pijpij

�¼ @tr


p@!mn

@

@pmn

�1

2pi

ipjj � pijpij

�

¼ 1

2ðffiffiffiffiffiffiffiffiffiffiffiffiffi��1!

q�1��1Þmnð!mnp

ii � 2pmnÞ

¼ 1

2pi


q� trð


qp!Þ; (B3)

where p represents a matrix with components pmn.In this case, CNL � 0 because there is an interaction between g and f, namely, a1 � 0. Thus, whether the Poisson

bracket is nontrivial or not is determined by the interaction pattern.


084052-9

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http://arXiv.org/abs/gr-qc/0405109








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When is multimetric gravity ghost-free?

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