Research ArticleConsistent Extension of Quasidilaton Massive Gravity

Josef Kluso^

Department of Theoretical Physics and Astrophysics Faculty of Science Masaryk University Kotlarska 2 611 37 Brno Czech Republic

Correspondence should be addressed to Josef Kluson kluphysicsmunicz

Received 22 May 2014 Accepted 4 July 2014 Published 20 July 2014

Academic Editor Sergey D Odintsov

Copyright copy 2014 Josef Kluson This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is devoted to the Hamiltonian analysis of extension of the quasidilaton massive gravity as was proposed recently in[arXiv13065502] We show that for given formulation of the theory the additional primary constraint that is responsible for theelimination of the Boulware-Deser ghost is missing We compare this situation with the quasidilaton massive gravity Finally wepropose the ghost-free extension of quasidilaton massive gravity

1 Introduction and Summary

Recently new version of the full nonlinear massive gravitywas found by de Rham Gabadadze and Toley (dRGT) [1 2]that provides the positive answer to the question of whethergraviton can have a nonzero mass In fact among manyremarkable properties there is the crucial one which is theabsence of the Boulware-Deser ghost [3 4]

The consistent massive gravity could also provide apossible explanation of the observed acceleration of thecosmic expansion which is one of the greatest mysteriesin modern cosmology It is tempting to speculate that thefinite graviton mass could be a source of the acceleratedexpansion of the universe For that reason it is of greatinterest to formulate theoretically consistent cosmologicalscenario in massive gravity that is also in agreement with theobservations Unfortunately it was recently shown that allhomogeneous and isotropic cosmological solutions in dRGTtheory are unstable [5] see also [6ndash8]

In order to resolve this problem we have two possibleoptions either to break homogeneity [9] or isotropy [10 11]or to extend the theory as in [12 13] Recently in [14] deFelice and Mukohyama proposed new extension of thequasidilaton massive gravity that could provide stable andself-accelerating homogeneous and isotropic cosmologicalsolution They further argued that given extension belongsto the class of models studied in [15] that are free from theBoulware-Deser ghosts However the explicit Hamiltoniananalysis of given theory was not performed in [14]

The goal of this paper is to reconsider the problem ofthe Boulware-Deser ghost in the model [14] We present anevidence that for the action that was introduced in [14] theBoulware-Deser ghost cannot be eliminated More preciselyperforming the Hamiltonian analysis of this model with thetime dependent quasidilaton we find that the primary con-straint which is responsible for the elimination of the ghostin Stuckelberg formulation of nonlinear massive gravity [16ndash19] is missing (for related work see [20]) This result impliesthat generally Boulware-Deser ghost is present On the otherhand we show that this additional constraint emerges in twocases when 120596 = 0 and when 120572

120590= 0 We also propose model

of consistent extension of the quasidilaton nonlinear massivegravity that can be considered as the generalization of thecoupling betweenmassive gravity and the galileon [21]Thenwe argue that given theory is ghost-free following [19]

This paper is organized as follows In next Section 2 wereview the basic facts about extension of the quasidilatonnonlinear massive gravity as was proposed in [14] Thenwe proceed to the Hamiltonian analysis of given theory andargue that there is no scalar primary constraint that couldeliminate the Boulware-Deser ghost We also discuss theanalysis performed recently in [22] andwe argue that the con-straints introduced there cannot be considered as the gaugefixing constraint In Section 3 we perform the Hamiltoniananalysis of quasidilaton nonlinear massive gravity when wefind that there is an additional primary constraintThis resultshows that the quasidilaton massive theory as was proposedin [12] is ghost-free at least in theirminimal version Finally in

Hindawi Publishing CorporationJournal of GravityVolume 2014 Article ID 413835 7 pageshttpdxdoiorg1011552014413835

2 Journal of Gravity

Section 4 we propose an extension of the quasidilaton mas-sive gravity that is ghost-free and that can be considered as thegeneralization of proposal [14] It would be extremely inter-esting to analyze cosmological consequences of this theory

2 Extension of Quasidilaton Massive Gravity

In this section we review basic facts about extension ofquasidilaton massive gravity as was proposed in [14] Forsimplicity we restrict ourselves to the minimal form of themassive gravity keeping in mind that its generalization isstraightforward

Explicitly let us consider the following action

119878 = 119878mg + 119878120590

119878mg = 1198722

119901int1198894

119909radicminus119892 [(4)

119877 + 21198982

(3 minus Ω (Φ)radic119892minus1 119891)]

(1)

where 119891120583] was introduced in [14]

119891120583] = 119891120583] minus

120572120590

1198722

1199011198982119890minus2120590119872119901

120597120583120590120597]120590 119891

120583] = 120597120583120601119886

120597]120601119887

120578119886119887

(2)

where 120601119886 119886 119887 = 0 1 2 3 are Stuckelberg fields and where120578119886119887= diag(minus1 1 1 1) Further 119878

120590is defined as

119878120590= minus

120596

2

int1198894

119909radicminus119892119892120583]120597120583120590120597]120590 (3)

Note that Ω(Φ) is a function of 120590 which is necessary for theinvariance of the theory under global symmetry

120590 997888rarr 120590 + 1205900 120601

119886

997888rarr 119890minus1205900119872119901

120601119886 (4)

so that under (4) 119891120583] and 119891120583] transform as

119891120583] 997888rarr 119890

minus21205900119872119901119891120583]

119891120583] 997888rarr 119890

minus21205900119872119901 119891120583] (5)

The massive term contains square root of the expression119892120583] 119891]120588 that under (4) transforms as

radic119892minus1 119891 997888rarr 119890

minus1205900119872119901radic119892minus1 119891 (6)

which implies thatΩ(120590) has to have the form

Ω (120590) = 119890120590119872119901

(7)

Now we are ready to proceed to the Hamiltonian analysis ofgiven theory Due to the presence of the square root in theaction we perform the redefinition of the shift functions [2324]

119873119894

= 119872119899119894

+119891119894119896 1198910119896+ 119873119863

119894

119895119899119895

(8)

where

1198722

= minus11989100+1198910119896

119891119896119897 1198911198970

119891119894119895

119891119895119896

= 120575119895

119894 (9)

and where119863119894119895obeys the equation

radic119909119863119894

119895= radic(119892

119894119896minus 119863119894

119898119899119898119863119896

119899119899119899)119891119896119895 119909 = 1 minus 119899

119894 119891119894119895119899119895 (10)

and also the following important identity

119891119894119896119863119896

119895=119891119895119896119863119896

119894 (11)

We also use 3 + 1 decomposition of the four-dimensionalmetric 119892

120583] [25 26]

11989200= minus119873

2

+ 119873119894119892119894119895

119873119895 119892

0119894= 119873119894 119892119894119895= 119892119894119895

11989200

= minus

1

1198732 119892

0119894

=

119873119894

1198732 119892

119894119895

= 119892119894119895

minus

119873119894

119873119895

1198732

(12)

Note that in 3 + 1 formalism the kinetic term for 120590 has theform

minus

120596

1198722

119901

119892120583]120597120583120590120597]120590 =

120596

1198722

119901

(nabla119899120590)2

minus

120596

1198722

119901

120597119894120590119892119894119895

120597119895120590 (13)

where after redefinition (8) nabla119899120590 has the form

nabla119899120590 =

1

119873

(120597119905120590 minus (119872119899

119894

+119891119894119896 1198910119896+ 119873119863

119894

119895119899119895

) 120597119894120590) (14)

With the help of these expressions we rewrite action (1) intothe form

119878 = 1198722

119901int1198893x119889119905 [119873radic119892

119894119895G119894119895119896119897

119896119897+ 119873radic119892119877 minus radic119892119872119880minus

minus 21198982

(119873Ω (Φ)radic119892radic119909119863119894

119894minus 3119873radic119892)+

+119873radic119892

120596

21198722

119901

(nabla119899120590)2

minus119873radic119892

120596

21198722

119901

120597119894120590119892119894119895

120597119895120590]

(15)

where

119880 = 21198982

Ω (Φ)radic119909 (16)

and where we used the 3 + 1 decomposition of the four-dimensional scalar curvature

(4)

119877 = 119894119895G119894119895119896119897

119896119897 + 119877 (17)

where 119877 is three-dimensional scalar curvature We alsointroduced de Witt metric

G119894119895119896119897

=

1

2

(119892119894119896

119892119895119897

+ 119892119894119897

119892119895119896

) minus 119892119894119895

119892119896119897 (18)

with inverse

G119894119895119896119897=

1

2

(119892119894119896119892119895119897+ 119892119894119897119892119895119896) minus

1

2

119892119894119895119892119896119897

G119894119895119896119897

G119896119897119898119899

=

1

2

(120575119898

119894120575119899

119895+ 120575119899

119894120575119898

119895)

(19)

Journal of Gravity 3

Note that in (17) we ignored the terms containing totalderivatives Finally note that

119894119895is defined as

119894119895=

1

2119873

(120597119905119892119894119895minus nabla119894119873119895(119899 119892) minus nabla

119895119873119894(119899 119892)) (20)

where119873119894depends on 119899119894 and 119892 through relation (8)

Now we could proceed to the Hamiltonian formulationof given theory However the structure of the derivative nabla

119899120590

(14) suggests very complicated relations between momentaand velocities For that reason we consider simpler casewhenwe presume that 120590 depends on time only Note that this isthe reasonable approximation that does not spoil the physicalcontent of the theory From (15) we find the momentaconjugate to119873 119899119894 and 119892

119894119895

120587119873asymp 0 120587

119894asymp 0 120587

119894119895

= 1198722

119901radic119892G119894119895119896119897

119896119897 (21)

While in case of 120601119886 and 120590 we find

119901119886=

M119886119887120597119905120601119887

119872

[119899119894

R119894+1198722

119901radic119892119880] minus 119891

119894119895

120597119895120601119886R119894

119901120590= minus

1

119872

120572120590

1198722

1199011198982119890minus2120590119872

2

119901120597119905120590 (119899119894

R119894+1198722

119901radic119892119880) + 120596radic119892120597

119905120590

(22)

where

1198722

= 1198722

0+

120572120590

1198722

1199011198982119890minus2120590119872

2

119901(120597119905120590)2

R119894= minus2119892

119894119896nabla119895120587119896119895

1198722

0= minus120597119905120601119886

M119886119887120597119905120601119887

M119886119887= 120578119886119887minus 120597119894120601119886119891119894119895

120597119895120601119887

(23)

These relations imply

1198722

0= minus

1

Π119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

Π119886M119886119887Π

119887120572120590

1198722

1199011198982

times 119890minus2120590119872

2

119901(120597119905120590)2

(24)

where Π119886= 119901119886+ 119891119894119895

120597119895120601119886R119894 Then it is easy to find relation

between momenta and velocities

119901120590+ radic

120572120590

1198722

1199011198982119890minus120590119872

2

119901radicΠ119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

=

1

119873

120596radic119892 120597119905 120590

times

Π119886

radicΠ119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

=

119872119901119898

radic120572120590

1198901205901198722

119901M119886119887

120597119905120601119887

120597119905120590

(25)

It is crucial that these relations do not imply an existence ofthe scalar primary constraint which is in sharp contrast with

the case of the dRGT massive gravity [16] or dRGT massivegravity coupled to the galileon [19] On the other hand usingthe property of the matrixM

119886119887 we find three constraints

120597119894120601119886

Π119886equiv Σ119894= 120597119894120601119886

119901119886+R119894asymp 0 (26)

With suitable extension of these constraints by terms propor-tional to the primary constraints 120587

119894asymp 0 we find that they

are the first class constraints whose smeared forms are thegenerator of spatial diffeomorphism

Now using (25) we determine corresponding Hamilto-nian

119867 = int1198893x119873H

0 (27)

where

H0=

1

2120596radic119892

(119901120590+ radic

120572120590

1198722

1199011198982119890minus120590119872

2

119901

times radicΠ119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

)

2

+

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus radic119892(3)

119877

minus 21198982

(Ωradic119892radic119909119863119894

119894minus 2radic119892) + 119863

119894

119895119899119895

R119894

(28)

The requirement of the preservation of the constraint 120587119873asymp

0 implies that H0is a constraint as well The succeeding

analysis is straightforward We have six first class constraintsΣ119894asymp 0 H

0asymp 0 120587

119873asymp 0 and six second class constraints

120587119894asymp 0 C

119894asymp 0 where C

119894asymp 0 are the secondary constraints

that arise from the requirement of the preservation of theconstraints 120587

119894asymp 0 These constraints can be solved for 120587

119894

and 119899119894 Further the first class constraint 120587119873asymp 0 can be gauge

fixed that leads to the elimination of 120587119873and119873 as dynamical

variables Finally the constraints Σ119894 H0can be again gauge

fixed which leads to the elimination of the Stuckelberg fieldsand conjugate momenta As a result we are left with 12degrees of freedom coming from the gravity sector that canbe identified with 10 degrees of freedom corresponding to themassive graviton and two degrees of freedom correspondingto the scalar at least at the linearized approximationNote thatthis scalar mode cannot be eliminated due to the absence ofthe scalar constraint As a result the Boulware-Deser ghost isgenerally present Finally there are two phase space degreesof freedom 120590 and 119901

120590

21 Remark about the Paper [22] After the first version ofthis paper was published Mukohyama argued in his paper[22] that the extension of the quasidilaton theory is ghost-free His arguments is based on the existence of the additionalconstraints that he imposed on the scalar fields

1206010

= minus119890minus120590119872119901

120601119894

= 120575119894

120583119909120583

119894 = 1 2 3 (29)

He called this constraint as the gauge fixing constraint andthen he was able to derive the Hamiltonian in the form

4 Journal of Gravity

119867 = int1198893x119873C

0 where the specific form of C

0is given in

[22] According to this result he claimed that there exists anadditional constraint in the gauge fixed theory so that thisconstraint is responsible for the elimination of the Boulware-Deser ghost

In this section we reconsider (29) from different point ofview and argue that this is not a gauge fixing constraint at allWe do not impose the fixing of spatial diffeomorphism andconsider the following relation between 120601

0and 120590 (note that

we do not use the word ldquogauge fixing conditionrdquo)

120590 = minus119872119901ln1206010 (30)

that is equivalent to (29) Inserting this relation to thedefinition of 119891

120583] we obtain

119891120583] = 120578119886119887120597120583120601

119886

120597]120601119887

minus

120572120590

1198982

119892

1205971205831206010

120597]1206010

(31)

Then inserting (31) into the dRGTmassive gravity we obtain

119878 = 1198722

119901int1198894

119909radicminus119892[(4)

119877 + 21198982

(3 minus

1

(1206010)2

radic119892minus1 119891)

minus

120596

2(1206010)2119892120583]1205971205831206010

120597]1206010

]

(32)

Now we see that given action is manifestly diffeomorphisminvariant In other words while we reduce the number ofdegrees of freedom by imposing (30) the number of gaugesymmetries is the same However we mean that the fact thatwe have less degrees of freedom than the original theorywhilethe number of gauge symmetries is the same implies thatthese two theories have different physical content and shouldnot be considered as equivalent In other words condition(29) is not a gauge fixing condition

From given analysis it is clear that the constraint C0

that was identified in [22] corresponds to the Hamiltonianconstraint in the theory with fixed spatial diffeomorphismClearly this constraint should have vanishing Poisson bracketC0(x)C

0(y) which implies that it is the first class con-

straint Clearly there is no way how to generate the additionalconstraint by imposing the requirement of the preservationof the constraint C

0during the time evolution of the system

since the Hamiltonian is equal to 119867 = int1198893x119873C0 However

sinceC0is the first class constraint it can be gauge fixed and

hence we reduce the number of physical degrees of freedomby two But we should again stress that the theory representedby action (33) is not equivalent to action (1) that representsthe extension of the quasidilaton dRGT theory In summarywe mean that the arguments that were presented in [22] donot prove that the extension of quasidilaton massive gravityis ghost-free

Let us compare this situation with the imposing staticgauge for the Hamiltonian (27) This gauge fixing is repre-sented by four gauge fixed functions

G0= 1206010

minus 119905 asymp 0 G119894= 120601119894

minus 119909119894

asymp 0 (33)

These constraints together withH0 Σ119894form the second class

constraints that can be explicitly solved for 119901119886 We solve H

0

for 1199010since we can identify the gauge fixed Hamiltonian

with 1199010= minusHgf The resulting Hamiltonian describes the

dynamic of the physical degrees of freedom 119892119894119895 120587119894119895 and 120590

119901120590 The detailed counting of the physical degrees of freedom

was presented in the end of previous section and we will notrepeat it here

3 The Case 120572120590= 0

In previous section we saw that the extension of the qua-sidilaton theory that was suggested in [14] suffers fromthe presence of Boulware-Deser ghost due to the absenceof two additional scalar constraints in the Hamiltonianformulation of given theory It is instructive to see whetherthese constraints emerge in the case of quasidilaton massivegravity which corresponds to the situation when 120572

120590= 0

Explicitly we consider the action [11 12]

119878 = 1198722

119901int1198893x119889119905 [119873radic119892

119894119895G119894119895119896119897

119896119897+ 119873radic119892119877 minus radic119892119872

0119880

minus 21198982

(119873radic119892radic119909Ω119863

119894

119894minus 3119873radic119892)

+ 119873radic119892

120596

1198722

119901

(nabla119899120590)2

minus 119873radic119892

120596

1198722

119901

120597119894120590119892119894119895

120597119895120590]

(34)

Let us now perform the Hamiltonian analysis of action (34)First of all we find the canonical momenta

119901119886= minus(minusM

119886119887

1

1198720

120597119905120601119887

119899119894

+ 119891119894119895

120597119895120601119886)R119894

+

1

1198720

1198722

119901radic119892M

119886119887120597119905120601119887

119880

+

1

1198720

M119886119887120597119905120601119887

119899119894

120597119894120590119901120590minus 120597119894120590119891119894119896

120597119896120601119886119901120590

119901120590= 120596radic119892nabla

119899120590

(35)

so that it is easy to find the scalar primary constraint

Σ119901equiv (119901119886+ (R119894+ 120597119894120590119901120590) 119891119894119896

120597119896120601119886) 120578119886119887

times (119901119887+ (R119894+ 120597119894120590119901120590) 119891119894119896

120597119896120601119887)

+ (119899119894

(R119894+ 119901120590120597119894120590) +119872

2

119901radic119892119880)

2

asymp 0

(36)

and also using the fact that 120597119894120601119886M119886119887= 0 we find additional

three constraints

Σ119894= 119901119886120597119894120601119886

+R119894+ 120597119894120590119901120590 (37)

Now using these constraints we can simplify (36) so that ithas the form

Σ119901= 119901119886M119886119887

119901119887+ (119899119894

(R119894+ 119901120590120597119894120590) +119872

2

119901radic119892119880)

2

asymp 0 (38)

Journal of Gravity 5

This has exactly the same form as the scalar constraint thatemerges in case of dRGT massive gravity written in theStuckelberg formalismThe minimal form of this gravity wasanalyzed in [16] and this analysis can be easily applied to ourcase From (15) we find the Hamiltonian with all primaryconstraints included

119867119864= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (39)

where we introduced the constraint

Σ119894= Σ119894+ 120597119894119899119895

120587119895+ 120597119895(119899119895

120587119894) (40)

and where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877 + 2119898

2

1198722

119901radic119892radic119909Ω119863

119894

119894

minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

(R119894+ 119901120590120597119894120590)

+

1

radic119892120596

1199012

120590+ 120596radic119892119892

119894119895

120597119894120590120597119895120590

(41)

As the next step we have to perform the analysis of thestability of the primary constraints120587

119894asymp 0120587

119873asymp 0 andΣ

119901asymp 0

In case of the constraint 120587119894asymp 0 we find

120597119905120587119894= 120587119894 119867 = minus[(R

119895+ 119901120590120597119895120590) minus

21198982

1198722

119901Ωradic119892

radic119909

119899119896

119891119896119895]

times [119873

120575 (119863119895

119898119899119898

)

120575119899119894

+ 120575119895

119894(119899119894

(R119894+ 119901120590120597119894120590)

+1198722

119901radic119892119880)] = 0

(42)

so that we impose the following secondary constraint

C119894=R119894+ 119901120590120597119894120590 minus

21198982

1198722

119901Ωradic119892

radic119909

119891119894119895119899119895

(43)

Now with the help of this constraint and the constraint Σ119894 we

can simplify Σ119901in a similar way as in [16]

Σ119901= 41198984

1198724

119901119892Ω2

+ 119901119860120578119860119861

119901119861 (44)

Then it is easy to show that Σ119901(x) Σ119901(y) = 0 and the

requirement of the preservation of given constraint leads tothe emergence of an additional constraint These constraintsare responsible for the elimination of Boulware-Deser ghostsee again [16] formore details In other words the presence ofthe kinetic term for the quasidilaton that minimally couplesto gravity does not spoil the property that the given theory isghost-free

4 Ghost-Free Extension of QuasidilatonMassive Gravity

We argued in Section 2 that the extension of quasidilatontheory as was formulated in [14] is plagued by the presenceof the Boulware-Deser ghost On the other hand giventheory has many nice properties so that it is desirable topropose its ghost-free version In this section we proposesuch a formulation when we replace the kinetic term for 120590by following tadpole galileon term

119878120590=minus119879int119889

4

119909Ψ (Φ119860

)radicminus det 119891120583] = minus119879int119889

4

119909Ψ (Φ119860

)119872radic119891

(45)

where Φ119860 = (120601119886 120590) 119891 = det 119891119894119895and where the function

Ψ(Φ119860

) was chosen in such a way that action (45) is invariantunder (4) We claim that the quasidilaton theory formulatedas the dRGT massive theory with 119891

120583] and with the kineticterm for the galileon given by (45) is ghost-free

To see this explicitly it is useful to introduce the followingnotation Let s write 119891

120583] as

119891120583] = 120597120583Φ

119860

G119860119861120597]Φ119861

(46)

where we introduced the metricG119860119861

G119860119861= (

120578119860119861

0

0 minus

120572120590

1198722

1199011198982119890minus2120590119872119901) (47)

We see that our proposal has the form of the galileon coupledto dRGT massive gravity [21] whose Hamiltonian analysiswas performed in [19] On the other hand the action definedby (45) is more complicated since the metric 120578

119860119861is replaced

with the more general metric G119860119861

that depends on Φ119860 andthere are also additional scalar fieldsΩ(120601119860)Ψ(Φ119860) Howeverwe can expect that this fact will not modify the constraintstructure of given theory

To see this explicitly let us briefly review theHamiltoniananalysis of the nonlinear massive gravity with the term (45)keeping in mind that more detailed analysis can be found in[19] As usual the momenta conjugate to119873 119899119894 and 119892

119894119895are

120587119873asymp 0 120587

119894asymp 0 120587

119894119895

= 1198722

119901radic119892G119894119895119896119897

119896119897

(48)

while the momentum conjugate to Φ119860 has the form

119901119860= minus(

120575119872

120575120597119905Φ119860119899119894

+G119860119861

119891119894119895

120597119895Φ119861

)R119894minus1198722

119901radic119892

120575119872

120597119905Φ1198601198801015840

(49)

where

1198801015840

= 21198982

Ω (Φ)radic119909 +

119879

1198722

119901

Ψ (Φ)radic119891 (50)

and where1198722 has the form

1198722

= minus120597119905Φ119860

M119860119861120597119905Φ119861

M119860119861= G119860119861minusG119860119862120597119894Φ119862 119891119894119895

120597119895Φ119863

G119863119861

(51)

6 Journal of Gravity

Note that the matrixM119860119861

obeys following relations

M119860119861G119861119862

M119862119863=M119860119863 120597

119894Φ119860

M119860119861= 0 (52)

Then it is easy to determine the following primary con-straints

Σ119901= (119899119894

R119894+1198722

119901radic1198921198801015840

)

2

+ (119901119860+R119894

119891119894119895

G119860119862120597119895Φ119862

)G119860119861

times (119901119861+R119894

119891119894119895

G119861119863120597119895Φ119863

) asymp 0

120597119894Φ119860

119901119860+R119894= Σ119894asymp 0

(53)

Now we are ready to write the extended Hamiltonian whichincludes all the primary constraints

119867119864= int1198893x (119873C

0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (54)

where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877

+ 21198982

1198722

119901radic119892Ω (Φ)

radic119909119863119894

119894minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

R119894

(55)

and where we introduced the constraints Σ119894defined as

Σ119894= Σ119894+ 120597119894119899119894

120587119894+ 120597119895(119899119895

120587119894) (56)

To proceed further we have to check the stability ofall constraints The procedure is the same as in [19] sothat we find that Σ

119894are the first class constraints while the

requirement of the preservation of the constraints 120587119894asymp 0

implies following secondary constraints [23 24]

C119894equivR119894minus

21198982

1198722

119901Ω (Φ)radic119892

radic119909

119891119894119895119899119895

asymp 0 (57)

Further the requirement of the preservation of the constraint120587119873asymp 0 implies an existence of the secondary constraintC

0asymp

0 Using the constraints C119894and Σ

119894 we replace the constraint

Σ119901by new independent constraint Σ

119901

Σ119901= 41198984

1198724

119901Ω2

119892 + 119901119860G119860119861

119901119861+ 2119879Ψradic

119891

times radic119901119860120597119894120601119860 119891119894119895120597119895120601119861119901119861+ 411989841198724

119901Ω2119892 + 1198792

Ψ2 119891 = 0

(58)

Then the total Hamiltonian with all constraints included hasthe form

119867119879= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894+ Γ119894

C119894)

(59)

Now we are ready to analyze the stability of all constraintsthat appear in (59) Again the analysis is the same as in [19]

with slight complication that now there are additional termsΨ(Φ) Ω(Φ) together with G

119860119861(Φ119860

) in the definition ofthe action However these terms are local functions of Φ119860so that they do not affect the result that Σ

119901(x) Σ119901(y) asymp

0 As a result the requirement of the preservation of theconstraint Σ

119901asymp 0 implies new constraint Σ119868119868

119901asymp 0 These two

constraints are the second class constraints that can be usedfor the elimination of the Boulware-Deser ghost mode and itsconjugate momenta

In this section we proposed an extension of the quasidila-ton massive gravity that is ghost-free This proposal can begeneralized in differentways either consider themost generalpotential of the dRGT massive gravity or more complicatedkinetic term for 120590 It would be also very interesting to analyzethe cosmological consequences of the model with action (45)following [14]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Grant agency of the Czechrepublic under the Grant P20112G028

References

[1] C de Rham and G Gabadadze ldquoGeneralization of the Fierz-Pauli actionrdquo Physical Review D vol 82 Article ID 0440202010

[2] C de Rham G Gabadadze and A J Tolley ldquoResummationof massive gravityrdquo Physical Review Letters vol 106 Article ID231101 2011

[3] D G Boulware and S Deser ldquoCan gravitation have a finiterangerdquo Physical Review D vol 6 no 12 pp 3368ndash3382 1972

[4] D G Boulware and S Desser ldquoInconsistency of finite rangegravitationrdquo Physics Letters B vol 40 no 2 pp 227ndash229 1972

[5] A de Felice A E Gumrukcuoglu and SMukohyama ldquoMassivegravity non-linear instability of the homogeneous and isotropicuniverserdquo Physical Review Letters vol 109 Article ID 1711012012

[6] K Koyama G Niz and G Tasinato ldquoThe self-acceleratinguniverse with vectors inmassive gravityrdquo Journal of High EnergyPhysics vol 2011 no 12 article 65 2011

[7] G Tasinato K Koyama and G Niz ldquoVector inst abilities andself-acceleration in the decoupling limit of massive gravityrdquoPhysical Review D vol 87 Article ID 064029 2013

[8] N Khosravi G Niz K Koyama and G Tasinato ldquoStabilityof the self-accelerating universe in massive gravityrdquo Journal ofCosmology and Astroparticle Physics vol 2013 no 8 p 44 2013

[9] G DrsquoAmico C de Rham S Dubovsky G Gabadadze DPirtskhalava and A J Tolley ldquoMassive cosmologiesrdquo PhysicalReview D vol 84 Article ID 124046 2011

[10] A E Gumrukcuoglu C Lin and S Mukohyama ldquoAnisotropicFriedmann-Robertson-Walker universe from nonlinear mas-sive gravityrdquoPhysics Letters B vol 717 no 4-5 pp 295ndash298 2012

Journal of Gravity 7

[11] A de Felice A E Gumrukcuoglu C Lin and S MukohyamaldquoNonlinear stability of cosmological solutions in massive grav-ityrdquo Journal of Cosmology andAstroparticle Physics vol 1305 no5 article 035 2013

[12] G DrsquoAmico G Gabadadze L Hui andD Pirtskhalava ldquoQuasi-dilaton theory and cosmologyrdquo Physical Review D vol 87Article ID 064037 2013

[13] Q-G Huang Y-S Piao and S-Y Zhou ldquoMass-varyingmassivegravityrdquo Physical Review D vol 86 Article ID 124014 2012

[14] A de Felice and S Mukohyama ldquoTowards consistent extensionof quasidilaton massive gravityrdquo Physics Letters B vol 728 pp622ndash625 2014

[15] G Gabadadze K Hinterbichler J Khoury D Pirtskhalavaand M Trodden ldquoCovariant master theory for novel Galileaninvariant models and massive gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 86 no 12Article ID 124004 2012

[16] J Kluson ldquoNote about Hamiltonian formalism for generalnonlinear massive gravity action in Stuckelberg formalismrdquoInternational Journal of Modern Physics A Particles and FieldsGravitation Cosmology vol 28 Article ID 1350160 16 pages2013

[17] J Kluson ldquoNon-linear massive gravity with additional primaryconstraint and absence of ghostsrdquo Physical Review D vol 86Article ID 044024 2012

[18] S F Hassan A Schmidt-May and M von Strauss ldquoProof ofconsistency of nonlinear massive gravity in the Stuckelbergformulationrdquo Physics Letters B vol 715 pp 355ndash339 2012

[19] J Kluson ldquoHamiltonian analysis of minimal massive gravitycoupled to Galileon tadpole termrdquo Journal of High EnergyPhysics vol 2013 article 80 2013

[20] Q G Huang K C Zhang and S Y Zhou ldquoGeneralizedmassive gravity in arbitrary dimensions and its Hamiltonianformulationrdquo Journal of Cosmology and Astroparticle Physicsvol 1308 article 050 2013

[21] M Andrews G Goon K Hinterbichler J Stokes and MTrodden ldquoMassive gravity coupled to Galileons is ghost-freerdquoPhysical Review Letters vol 111 no 6 Article ID 061107 2013

[22] SMukohyama ldquoExtended quasidilatonmassive gravity is ghostfreerdquo httparxivorgabs13092146

[23] S FHassan andRA Rosen ldquoOnnon-linear actions formassivegravityrdquo Journal of High Energy Physics vol 2011 article 9 2011

[24] S F Hassan and R A Rosen ldquoResolving the ghost problemin non-linear massive gravityrdquo Physical Review Letters vol 108Article ID 041101 2012

[25] E Gourgoulhon ldquo3 + 1 formalism and bases of numericalrelativityrdquo httparxivorgabsgr-qc0703035

[26] R L Arnowitt S Deser and C W Misner ldquoThe dynamics ofgeneral relativityrdquo httparxivorgabsgrqc0405109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

2 Journal of Gravity

Section 4 we propose an extension of the quasidilaton mas-sive gravity that is ghost-free and that can be considered as thegeneralization of proposal [14] It would be extremely inter-esting to analyze cosmological consequences of this theory

2 Extension of Quasidilaton Massive Gravity

In this section we review basic facts about extension ofquasidilaton massive gravity as was proposed in [14] Forsimplicity we restrict ourselves to the minimal form of themassive gravity keeping in mind that its generalization isstraightforward

Explicitly let us consider the following action

119878 = 119878mg + 119878120590

119878mg = 1198722

119901int1198894

119909radicminus119892 [(4)

119877 + 21198982

(3 minus Ω (Φ)radic119892minus1 119891)]

(1)

where 119891120583] was introduced in [14]

119891120583] = 119891120583] minus

120572120590

1198722

1199011198982119890minus2120590119872119901

120597120583120590120597]120590 119891

120583] = 120597120583120601119886

120597]120601119887

120578119886119887

(2)

where 120601119886 119886 119887 = 0 1 2 3 are Stuckelberg fields and where120578119886119887= diag(minus1 1 1 1) Further 119878

120590is defined as

119878120590= minus

120596

2

int1198894

119909radicminus119892119892120583]120597120583120590120597]120590 (3)

Note that Ω(Φ) is a function of 120590 which is necessary for theinvariance of the theory under global symmetry

120590 997888rarr 120590 + 1205900 120601

119886

997888rarr 119890minus1205900119872119901

120601119886 (4)

so that under (4) 119891120583] and 119891120583] transform as

119891120583] 997888rarr 119890

minus21205900119872119901119891120583]

119891120583] 997888rarr 119890

minus21205900119872119901 119891120583] (5)

The massive term contains square root of the expression119892120583] 119891]120588 that under (4) transforms as

radic119892minus1 119891 997888rarr 119890

minus1205900119872119901radic119892minus1 119891 (6)

which implies thatΩ(120590) has to have the form

Ω (120590) = 119890120590119872119901

(7)

Now we are ready to proceed to the Hamiltonian analysis ofgiven theory Due to the presence of the square root in theaction we perform the redefinition of the shift functions [2324]

119873119894

= 119872119899119894

+119891119894119896 1198910119896+ 119873119863

119894

119895119899119895

(8)

where

1198722

= minus11989100+1198910119896

119891119896119897 1198911198970

119891119894119895

119891119895119896

= 120575119895

119894 (9)

and where119863119894119895obeys the equation

radic119909119863119894

119895= radic(119892

119894119896minus 119863119894

119898119899119898119863119896

119899119899119899)119891119896119895 119909 = 1 minus 119899

119894 119891119894119895119899119895 (10)

and also the following important identity

119891119894119896119863119896

119895=119891119895119896119863119896

119894 (11)

We also use 3 + 1 decomposition of the four-dimensionalmetric 119892

120583] [25 26]

11989200= minus119873

2

+ 119873119894119892119894119895

119873119895 119892

0119894= 119873119894 119892119894119895= 119892119894119895

11989200

= minus

1

1198732 119892

0119894

=

119873119894

1198732 119892

119894119895

= 119892119894119895

minus

119873119894

119873119895

1198732

(12)

Note that in 3 + 1 formalism the kinetic term for 120590 has theform

minus

120596

1198722

119901

119892120583]120597120583120590120597]120590 =

120596

1198722

119901

(nabla119899120590)2

minus

120596

1198722

119901

120597119894120590119892119894119895

120597119895120590 (13)

where after redefinition (8) nabla119899120590 has the form

nabla119899120590 =

1

119873

(120597119905120590 minus (119872119899

119894

+119891119894119896 1198910119896+ 119873119863

119894

119895119899119895

) 120597119894120590) (14)

With the help of these expressions we rewrite action (1) intothe form

119878 = 1198722

119901int1198893x119889119905 [119873radic119892

119894119895G119894119895119896119897

119896119897+ 119873radic119892119877 minus radic119892119872119880minus

minus 21198982

(119873Ω (Φ)radic119892radic119909119863119894

119894minus 3119873radic119892)+

+119873radic119892

120596

21198722

119901

(nabla119899120590)2

minus119873radic119892

120596

21198722

119901

120597119894120590119892119894119895

120597119895120590]

(15)

where

119880 = 21198982

Ω (Φ)radic119909 (16)

and where we used the 3 + 1 decomposition of the four-dimensional scalar curvature

(4)

119877 = 119894119895G119894119895119896119897

119896119897 + 119877 (17)

where 119877 is three-dimensional scalar curvature We alsointroduced de Witt metric

G119894119895119896119897

=

1

2

(119892119894119896

119892119895119897

+ 119892119894119897

119892119895119896

) minus 119892119894119895

119892119896119897 (18)

with inverse

G119894119895119896119897=

1

2

(119892119894119896119892119895119897+ 119892119894119897119892119895119896) minus

1

2

119892119894119895119892119896119897

G119894119895119896119897

G119896119897119898119899

=

1

2

(120575119898

119894120575119899

119895+ 120575119899

119894120575119898

119895)

(19)

Journal of Gravity 3

Note that in (17) we ignored the terms containing totalderivatives Finally note that

119894119895is defined as

119894119895=

1

2119873

(120597119905119892119894119895minus nabla119894119873119895(119899 119892) minus nabla

119895119873119894(119899 119892)) (20)

where119873119894depends on 119899119894 and 119892 through relation (8)

Now we could proceed to the Hamiltonian formulationof given theory However the structure of the derivative nabla

119899120590

(14) suggests very complicated relations between momentaand velocities For that reason we consider simpler casewhenwe presume that 120590 depends on time only Note that this isthe reasonable approximation that does not spoil the physicalcontent of the theory From (15) we find the momentaconjugate to119873 119899119894 and 119892

119894119895

120587119873asymp 0 120587

119894asymp 0 120587

119894119895

= 1198722

119901radic119892G119894119895119896119897

119896119897 (21)

While in case of 120601119886 and 120590 we find

119901119886=

M119886119887120597119905120601119887

119872

[119899119894

R119894+1198722

119901radic119892119880] minus 119891

119894119895

120597119895120601119886R119894

119901120590= minus

1

119872

120572120590

1198722

1199011198982119890minus2120590119872

2

119901120597119905120590 (119899119894

R119894+1198722

119901radic119892119880) + 120596radic119892120597

119905120590

(22)

where

1198722

= 1198722

0+

120572120590

1198722

1199011198982119890minus2120590119872

2

119901(120597119905120590)2

R119894= minus2119892

119894119896nabla119895120587119896119895

1198722

0= minus120597119905120601119886

M119886119887120597119905120601119887

M119886119887= 120578119886119887minus 120597119894120601119886119891119894119895

120597119895120601119887

(23)

These relations imply

1198722

0= minus

1

Π119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

Π119886M119886119887Π

119887120572120590

1198722

1199011198982

times 119890minus2120590119872

2

119901(120597119905120590)2

(24)

where Π119886= 119901119886+ 119891119894119895

120597119895120601119886R119894 Then it is easy to find relation

between momenta and velocities

119901120590+ radic

120572120590

1198722

1199011198982119890minus120590119872

2

119901radicΠ119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

=

1

119873

120596radic119892 120597119905 120590

times

Π119886

radicΠ119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

=

119872119901119898

radic120572120590

1198901205901198722

119901M119886119887

120597119905120601119887

120597119905120590

(25)

It is crucial that these relations do not imply an existence ofthe scalar primary constraint which is in sharp contrast with

the case of the dRGT massive gravity [16] or dRGT massivegravity coupled to the galileon [19] On the other hand usingthe property of the matrixM

119886119887 we find three constraints

120597119894120601119886

Π119886equiv Σ119894= 120597119894120601119886

119901119886+R119894asymp 0 (26)

With suitable extension of these constraints by terms propor-tional to the primary constraints 120587

119894asymp 0 we find that they

are the first class constraints whose smeared forms are thegenerator of spatial diffeomorphism

Now using (25) we determine corresponding Hamilto-nian

119867 = int1198893x119873H

0 (27)

where

H0=

1

2120596radic119892

(119901120590+ radic

120572120590

1198722

1199011198982119890minus120590119872

2

119901

times radicΠ119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

)

2

+

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus radic119892(3)

119877

minus 21198982

(Ωradic119892radic119909119863119894

119894minus 2radic119892) + 119863

119894

119895119899119895

R119894

(28)

The requirement of the preservation of the constraint 120587119873asymp

0 implies that H0is a constraint as well The succeeding

analysis is straightforward We have six first class constraintsΣ119894asymp 0 H

0asymp 0 120587

119873asymp 0 and six second class constraints

120587119894asymp 0 C

119894asymp 0 where C

119894asymp 0 are the secondary constraints

that arise from the requirement of the preservation of theconstraints 120587

119894asymp 0 These constraints can be solved for 120587

119894

and 119899119894 Further the first class constraint 120587119873asymp 0 can be gauge

fixed that leads to the elimination of 120587119873and119873 as dynamical

variables Finally the constraints Σ119894 H0can be again gauge

fixed which leads to the elimination of the Stuckelberg fieldsand conjugate momenta As a result we are left with 12degrees of freedom coming from the gravity sector that canbe identified with 10 degrees of freedom corresponding to themassive graviton and two degrees of freedom correspondingto the scalar at least at the linearized approximationNote thatthis scalar mode cannot be eliminated due to the absence ofthe scalar constraint As a result the Boulware-Deser ghost isgenerally present Finally there are two phase space degreesof freedom 120590 and 119901

120590

21 Remark about the Paper [22] After the first version ofthis paper was published Mukohyama argued in his paper[22] that the extension of the quasidilaton theory is ghost-free His arguments is based on the existence of the additionalconstraints that he imposed on the scalar fields

1206010

= minus119890minus120590119872119901

120601119894

= 120575119894

120583119909120583

119894 = 1 2 3 (29)

He called this constraint as the gauge fixing constraint andthen he was able to derive the Hamiltonian in the form

4 Journal of Gravity

119867 = int1198893x119873C

0 where the specific form of C

0is given in

[22] According to this result he claimed that there exists anadditional constraint in the gauge fixed theory so that thisconstraint is responsible for the elimination of the Boulware-Deser ghost

In this section we reconsider (29) from different point ofview and argue that this is not a gauge fixing constraint at allWe do not impose the fixing of spatial diffeomorphism andconsider the following relation between 120601

0and 120590 (note that

we do not use the word ldquogauge fixing conditionrdquo)

120590 = minus119872119901ln1206010 (30)

that is equivalent to (29) Inserting this relation to thedefinition of 119891

120583] we obtain

119891120583] = 120578119886119887120597120583120601

119886

120597]120601119887

minus

120572120590

1198982

119892

1205971205831206010

120597]1206010

(31)

Then inserting (31) into the dRGTmassive gravity we obtain

119878 = 1198722

119901int1198894

119909radicminus119892[(4)

119877 + 21198982

(3 minus

1

(1206010)2

radic119892minus1 119891)

minus

120596

2(1206010)2119892120583]1205971205831206010

120597]1206010

]

(32)

Now we see that given action is manifestly diffeomorphisminvariant In other words while we reduce the number ofdegrees of freedom by imposing (30) the number of gaugesymmetries is the same However we mean that the fact thatwe have less degrees of freedom than the original theorywhilethe number of gauge symmetries is the same implies thatthese two theories have different physical content and shouldnot be considered as equivalent In other words condition(29) is not a gauge fixing condition

From given analysis it is clear that the constraint C0

that was identified in [22] corresponds to the Hamiltonianconstraint in the theory with fixed spatial diffeomorphismClearly this constraint should have vanishing Poisson bracketC0(x)C

0(y) which implies that it is the first class con-

straint Clearly there is no way how to generate the additionalconstraint by imposing the requirement of the preservationof the constraint C

0during the time evolution of the system

since the Hamiltonian is equal to 119867 = int1198893x119873C0 However

sinceC0is the first class constraint it can be gauge fixed and

hence we reduce the number of physical degrees of freedomby two But we should again stress that the theory representedby action (33) is not equivalent to action (1) that representsthe extension of the quasidilaton dRGT theory In summarywe mean that the arguments that were presented in [22] donot prove that the extension of quasidilaton massive gravityis ghost-free

Let us compare this situation with the imposing staticgauge for the Hamiltonian (27) This gauge fixing is repre-sented by four gauge fixed functions

G0= 1206010

minus 119905 asymp 0 G119894= 120601119894

minus 119909119894

asymp 0 (33)

These constraints together withH0 Σ119894form the second class

constraints that can be explicitly solved for 119901119886 We solve H

0

for 1199010since we can identify the gauge fixed Hamiltonian

with 1199010= minusHgf The resulting Hamiltonian describes the

dynamic of the physical degrees of freedom 119892119894119895 120587119894119895 and 120590

119901120590 The detailed counting of the physical degrees of freedom

was presented in the end of previous section and we will notrepeat it here

3 The Case 120572120590= 0

In previous section we saw that the extension of the qua-sidilaton theory that was suggested in [14] suffers fromthe presence of Boulware-Deser ghost due to the absenceof two additional scalar constraints in the Hamiltonianformulation of given theory It is instructive to see whetherthese constraints emerge in the case of quasidilaton massivegravity which corresponds to the situation when 120572

120590= 0

Explicitly we consider the action [11 12]

119878 = 1198722

119901int1198893x119889119905 [119873radic119892

119894119895G119894119895119896119897

119896119897+ 119873radic119892119877 minus radic119892119872

0119880

minus 21198982

(119873radic119892radic119909Ω119863

119894

119894minus 3119873radic119892)

+ 119873radic119892

120596

1198722

119901

(nabla119899120590)2

minus 119873radic119892

120596

1198722

119901

120597119894120590119892119894119895

120597119895120590]

(34)

Let us now perform the Hamiltonian analysis of action (34)First of all we find the canonical momenta

119901119886= minus(minusM

119886119887

1

1198720

120597119905120601119887

119899119894

+ 119891119894119895

120597119895120601119886)R119894

+

1

1198720

1198722

119901radic119892M

119886119887120597119905120601119887

119880

+

1

1198720

M119886119887120597119905120601119887

119899119894

120597119894120590119901120590minus 120597119894120590119891119894119896

120597119896120601119886119901120590

119901120590= 120596radic119892nabla

119899120590

(35)

so that it is easy to find the scalar primary constraint

Σ119901equiv (119901119886+ (R119894+ 120597119894120590119901120590) 119891119894119896

120597119896120601119886) 120578119886119887

times (119901119887+ (R119894+ 120597119894120590119901120590) 119891119894119896

120597119896120601119887)

+ (119899119894

(R119894+ 119901120590120597119894120590) +119872

2

119901radic119892119880)

2

asymp 0

(36)

and also using the fact that 120597119894120601119886M119886119887= 0 we find additional

three constraints

Σ119894= 119901119886120597119894120601119886

+R119894+ 120597119894120590119901120590 (37)

Now using these constraints we can simplify (36) so that ithas the form

Σ119901= 119901119886M119886119887

119901119887+ (119899119894

(R119894+ 119901120590120597119894120590) +119872

2

119901radic119892119880)

2

asymp 0 (38)

Journal of Gravity 5

This has exactly the same form as the scalar constraint thatemerges in case of dRGT massive gravity written in theStuckelberg formalismThe minimal form of this gravity wasanalyzed in [16] and this analysis can be easily applied to ourcase From (15) we find the Hamiltonian with all primaryconstraints included

119867119864= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (39)

where we introduced the constraint

Σ119894= Σ119894+ 120597119894119899119895

120587119895+ 120597119895(119899119895

120587119894) (40)

and where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877 + 2119898

2

1198722

119901radic119892radic119909Ω119863

119894

119894

minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

(R119894+ 119901120590120597119894120590)

+

1

radic119892120596

1199012

120590+ 120596radic119892119892

119894119895

120597119894120590120597119895120590

(41)

As the next step we have to perform the analysis of thestability of the primary constraints120587

119894asymp 0120587

119873asymp 0 andΣ

119901asymp 0

In case of the constraint 120587119894asymp 0 we find

120597119905120587119894= 120587119894 119867 = minus[(R

119895+ 119901120590120597119895120590) minus

21198982

1198722

119901Ωradic119892

radic119909

119899119896

119891119896119895]

times [119873

120575 (119863119895

119898119899119898

)

120575119899119894

+ 120575119895

119894(119899119894

(R119894+ 119901120590120597119894120590)

+1198722

119901radic119892119880)] = 0

(42)

so that we impose the following secondary constraint

C119894=R119894+ 119901120590120597119894120590 minus

21198982

1198722

119901Ωradic119892

radic119909

119891119894119895119899119895

(43)

Now with the help of this constraint and the constraint Σ119894 we

can simplify Σ119901in a similar way as in [16]

Σ119901= 41198984

1198724

119901119892Ω2

+ 119901119860120578119860119861

119901119861 (44)

Then it is easy to show that Σ119901(x) Σ119901(y) = 0 and the

requirement of the preservation of given constraint leads tothe emergence of an additional constraint These constraintsare responsible for the elimination of Boulware-Deser ghostsee again [16] formore details In other words the presence ofthe kinetic term for the quasidilaton that minimally couplesto gravity does not spoil the property that the given theory isghost-free

4 Ghost-Free Extension of QuasidilatonMassive Gravity

We argued in Section 2 that the extension of quasidilatontheory as was formulated in [14] is plagued by the presenceof the Boulware-Deser ghost On the other hand giventheory has many nice properties so that it is desirable topropose its ghost-free version In this section we proposesuch a formulation when we replace the kinetic term for 120590by following tadpole galileon term

119878120590=minus119879int119889

4

119909Ψ (Φ119860

)radicminus det 119891120583] = minus119879int119889

4

119909Ψ (Φ119860

)119872radic119891

(45)

where Φ119860 = (120601119886 120590) 119891 = det 119891119894119895and where the function

Ψ(Φ119860

) was chosen in such a way that action (45) is invariantunder (4) We claim that the quasidilaton theory formulatedas the dRGT massive theory with 119891

120583] and with the kineticterm for the galileon given by (45) is ghost-free

To see this explicitly it is useful to introduce the followingnotation Let s write 119891

120583] as

119891120583] = 120597120583Φ

119860

G119860119861120597]Φ119861

(46)

where we introduced the metricG119860119861

G119860119861= (

120578119860119861

0

0 minus

120572120590

1198722

1199011198982119890minus2120590119872119901) (47)

We see that our proposal has the form of the galileon coupledto dRGT massive gravity [21] whose Hamiltonian analysiswas performed in [19] On the other hand the action definedby (45) is more complicated since the metric 120578

119860119861is replaced

with the more general metric G119860119861

that depends on Φ119860 andthere are also additional scalar fieldsΩ(120601119860)Ψ(Φ119860) Howeverwe can expect that this fact will not modify the constraintstructure of given theory

To see this explicitly let us briefly review theHamiltoniananalysis of the nonlinear massive gravity with the term (45)keeping in mind that more detailed analysis can be found in[19] As usual the momenta conjugate to119873 119899119894 and 119892

119894119895are

120587119873asymp 0 120587

119894asymp 0 120587

119894119895

= 1198722

119901radic119892G119894119895119896119897

119896119897

(48)

while the momentum conjugate to Φ119860 has the form

119901119860= minus(

120575119872

120575120597119905Φ119860119899119894

+G119860119861

119891119894119895

120597119895Φ119861

)R119894minus1198722

119901radic119892

120575119872

120597119905Φ1198601198801015840

(49)

where

1198801015840

= 21198982

Ω (Φ)radic119909 +

119879

1198722

119901

Ψ (Φ)radic119891 (50)

and where1198722 has the form

1198722

= minus120597119905Φ119860

M119860119861120597119905Φ119861

M119860119861= G119860119861minusG119860119862120597119894Φ119862 119891119894119895

120597119895Φ119863

G119863119861

(51)

6 Journal of Gravity

Note that the matrixM119860119861

obeys following relations

M119860119861G119861119862

M119862119863=M119860119863 120597

119894Φ119860

M119860119861= 0 (52)

Then it is easy to determine the following primary con-straints

Σ119901= (119899119894

R119894+1198722

119901radic1198921198801015840

)

2

+ (119901119860+R119894

119891119894119895

G119860119862120597119895Φ119862

)G119860119861

times (119901119861+R119894

119891119894119895

G119861119863120597119895Φ119863

) asymp 0

120597119894Φ119860

119901119860+R119894= Σ119894asymp 0

(53)

Now we are ready to write the extended Hamiltonian whichincludes all the primary constraints

119867119864= int1198893x (119873C

0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (54)

where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877

+ 21198982

1198722

119901radic119892Ω (Φ)

radic119909119863119894

119894minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

R119894

(55)

and where we introduced the constraints Σ119894defined as

Σ119894= Σ119894+ 120597119894119899119894

120587119894+ 120597119895(119899119895

120587119894) (56)

To proceed further we have to check the stability ofall constraints The procedure is the same as in [19] sothat we find that Σ

119894are the first class constraints while the

requirement of the preservation of the constraints 120587119894asymp 0

implies following secondary constraints [23 24]

C119894equivR119894minus

21198982

1198722

119901Ω (Φ)radic119892

radic119909

119891119894119895119899119895

asymp 0 (57)

Further the requirement of the preservation of the constraint120587119873asymp 0 implies an existence of the secondary constraintC

0asymp

0 Using the constraints C119894and Σ

119894 we replace the constraint

Σ119901by new independent constraint Σ

119901

Σ119901= 41198984

1198724

119901Ω2

119892 + 119901119860G119860119861

119901119861+ 2119879Ψradic

119891

times radic119901119860120597119894120601119860 119891119894119895120597119895120601119861119901119861+ 411989841198724

119901Ω2119892 + 1198792

Ψ2 119891 = 0

(58)

Then the total Hamiltonian with all constraints included hasthe form

119867119879= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894+ Γ119894

C119894)

(59)

Now we are ready to analyze the stability of all constraintsthat appear in (59) Again the analysis is the same as in [19]

with slight complication that now there are additional termsΨ(Φ) Ω(Φ) together with G

119860119861(Φ119860

) in the definition ofthe action However these terms are local functions of Φ119860so that they do not affect the result that Σ

119901(x) Σ119901(y) asymp

0 As a result the requirement of the preservation of theconstraint Σ

119901asymp 0 implies new constraint Σ119868119868

119901asymp 0 These two

constraints are the second class constraints that can be usedfor the elimination of the Boulware-Deser ghost mode and itsconjugate momenta

In this section we proposed an extension of the quasidila-ton massive gravity that is ghost-free This proposal can begeneralized in differentways either consider themost generalpotential of the dRGT massive gravity or more complicatedkinetic term for 120590 It would be also very interesting to analyzethe cosmological consequences of the model with action (45)following [14]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Grant agency of the Czechrepublic under the Grant P20112G028

References

[1] C de Rham and G Gabadadze ldquoGeneralization of the Fierz-Pauli actionrdquo Physical Review D vol 82 Article ID 0440202010

[2] C de Rham G Gabadadze and A J Tolley ldquoResummationof massive gravityrdquo Physical Review Letters vol 106 Article ID231101 2011

[3] D G Boulware and S Deser ldquoCan gravitation have a finiterangerdquo Physical Review D vol 6 no 12 pp 3368ndash3382 1972

[4] D G Boulware and S Desser ldquoInconsistency of finite rangegravitationrdquo Physics Letters B vol 40 no 2 pp 227ndash229 1972

[5] A de Felice A E Gumrukcuoglu and SMukohyama ldquoMassivegravity non-linear instability of the homogeneous and isotropicuniverserdquo Physical Review Letters vol 109 Article ID 1711012012

[6] K Koyama G Niz and G Tasinato ldquoThe self-acceleratinguniverse with vectors inmassive gravityrdquo Journal of High EnergyPhysics vol 2011 no 12 article 65 2011

[7] G Tasinato K Koyama and G Niz ldquoVector inst abilities andself-acceleration in the decoupling limit of massive gravityrdquoPhysical Review D vol 87 Article ID 064029 2013

[8] N Khosravi G Niz K Koyama and G Tasinato ldquoStabilityof the self-accelerating universe in massive gravityrdquo Journal ofCosmology and Astroparticle Physics vol 2013 no 8 p 44 2013

[9] G DrsquoAmico C de Rham S Dubovsky G Gabadadze DPirtskhalava and A J Tolley ldquoMassive cosmologiesrdquo PhysicalReview D vol 84 Article ID 124046 2011

[10] A E Gumrukcuoglu C Lin and S Mukohyama ldquoAnisotropicFriedmann-Robertson-Walker universe from nonlinear mas-sive gravityrdquoPhysics Letters B vol 717 no 4-5 pp 295ndash298 2012

Journal of Gravity 7

[11] A de Felice A E Gumrukcuoglu C Lin and S MukohyamaldquoNonlinear stability of cosmological solutions in massive grav-ityrdquo Journal of Cosmology andAstroparticle Physics vol 1305 no5 article 035 2013

[12] G DrsquoAmico G Gabadadze L Hui andD Pirtskhalava ldquoQuasi-dilaton theory and cosmologyrdquo Physical Review D vol 87Article ID 064037 2013

[13] Q-G Huang Y-S Piao and S-Y Zhou ldquoMass-varyingmassivegravityrdquo Physical Review D vol 86 Article ID 124014 2012

[14] A de Felice and S Mukohyama ldquoTowards consistent extensionof quasidilaton massive gravityrdquo Physics Letters B vol 728 pp622ndash625 2014

[15] G Gabadadze K Hinterbichler J Khoury D Pirtskhalavaand M Trodden ldquoCovariant master theory for novel Galileaninvariant models and massive gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 86 no 12Article ID 124004 2012

[16] J Kluson ldquoNote about Hamiltonian formalism for generalnonlinear massive gravity action in Stuckelberg formalismrdquoInternational Journal of Modern Physics A Particles and FieldsGravitation Cosmology vol 28 Article ID 1350160 16 pages2013

[17] J Kluson ldquoNon-linear massive gravity with additional primaryconstraint and absence of ghostsrdquo Physical Review D vol 86Article ID 044024 2012

[18] S F Hassan A Schmidt-May and M von Strauss ldquoProof ofconsistency of nonlinear massive gravity in the Stuckelbergformulationrdquo Physics Letters B vol 715 pp 355ndash339 2012

[19] J Kluson ldquoHamiltonian analysis of minimal massive gravitycoupled to Galileon tadpole termrdquo Journal of High EnergyPhysics vol 2013 article 80 2013

[20] Q G Huang K C Zhang and S Y Zhou ldquoGeneralizedmassive gravity in arbitrary dimensions and its Hamiltonianformulationrdquo Journal of Cosmology and Astroparticle Physicsvol 1308 article 050 2013

[21] M Andrews G Goon K Hinterbichler J Stokes and MTrodden ldquoMassive gravity coupled to Galileons is ghost-freerdquoPhysical Review Letters vol 111 no 6 Article ID 061107 2013

[22] SMukohyama ldquoExtended quasidilatonmassive gravity is ghostfreerdquo httparxivorgabs13092146

[23] S FHassan andRA Rosen ldquoOnnon-linear actions formassivegravityrdquo Journal of High Energy Physics vol 2011 article 9 2011

[24] S F Hassan and R A Rosen ldquoResolving the ghost problemin non-linear massive gravityrdquo Physical Review Letters vol 108Article ID 041101 2012

[25] E Gourgoulhon ldquo3 + 1 formalism and bases of numericalrelativityrdquo httparxivorgabsgr-qc0703035

[26] R L Arnowitt S Deser and C W Misner ldquoThe dynamics ofgeneral relativityrdquo httparxivorgabsgrqc0405109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Gravity 3

Note that in (17) we ignored the terms containing totalderivatives Finally note that

119894119895is defined as

119894119895=

1

2119873

(120597119905119892119894119895minus nabla119894119873119895(119899 119892) minus nabla

119895119873119894(119899 119892)) (20)

where119873119894depends on 119899119894 and 119892 through relation (8)

Now we could proceed to the Hamiltonian formulationof given theory However the structure of the derivative nabla

119899120590

(14) suggests very complicated relations between momentaand velocities For that reason we consider simpler casewhenwe presume that 120590 depends on time only Note that this isthe reasonable approximation that does not spoil the physicalcontent of the theory From (15) we find the momentaconjugate to119873 119899119894 and 119892

119894119895

120587119873asymp 0 120587

119894asymp 0 120587

119894119895

= 1198722

119901radic119892G119894119895119896119897

119896119897 (21)

While in case of 120601119886 and 120590 we find

119901119886=

M119886119887120597119905120601119887

119872

[119899119894

R119894+1198722

119901radic119892119880] minus 119891

119894119895

120597119895120601119886R119894

119901120590= minus

1

119872

120572120590

1198722

1199011198982119890minus2120590119872

2

119901120597119905120590 (119899119894

R119894+1198722

119901radic119892119880) + 120596radic119892120597

119905120590

(22)

where

1198722

= 1198722

0+

120572120590

1198722

1199011198982119890minus2120590119872

2

119901(120597119905120590)2

R119894= minus2119892

119894119896nabla119895120587119896119895

1198722

0= minus120597119905120601119886

M119886119887120597119905120601119887

M119886119887= 120578119886119887minus 120597119894120601119886119891119894119895

120597119895120601119887

(23)

These relations imply

1198722

0= minus

1

Π119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

Π119886M119886119887Π

119887120572120590

1198722

1199011198982

times 119890minus2120590119872

2

119901(120597119905120590)2

(24)

where Π119886= 119901119886+ 119891119894119895

120597119895120601119886R119894 Then it is easy to find relation

between momenta and velocities

119901120590+ radic

120572120590

1198722

1199011198982119890minus120590119872

2

119901radicΠ119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

=

1

119873

120596radic119892 120597119905 120590

times

Π119886

radicΠ119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

=

119872119901119898

radic120572120590

1198901205901198722

119901M119886119887

120597119905120601119887

120597119905120590

(25)

It is crucial that these relations do not imply an existence ofthe scalar primary constraint which is in sharp contrast with

the case of the dRGT massive gravity [16] or dRGT massivegravity coupled to the galileon [19] On the other hand usingthe property of the matrixM

119886119887 we find three constraints

120597119894120601119886

Π119886equiv Σ119894= 120597119894120601119886

119901119886+R119894asymp 0 (26)

With suitable extension of these constraints by terms propor-tional to the primary constraints 120587

119894asymp 0 we find that they

are the first class constraints whose smeared forms are thegenerator of spatial diffeomorphism

Now using (25) we determine corresponding Hamilto-nian

119867 = int1198893x119873H

0 (27)

where

H0=

1

2120596radic119892

(119901120590+ radic

120572120590

1198722

1199011198982119890minus120590119872

2

119901

times radicΠ119886M119886119887Π

119887+ (119899119894R119894+1198722

119901radic119892119880)

2

)

2

+

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus radic119892(3)

119877

minus 21198982

(Ωradic119892radic119909119863119894

119894minus 2radic119892) + 119863

119894

119895119899119895

R119894

(28)

The requirement of the preservation of the constraint 120587119873asymp

0 implies that H0is a constraint as well The succeeding

analysis is straightforward We have six first class constraintsΣ119894asymp 0 H

0asymp 0 120587

119873asymp 0 and six second class constraints

120587119894asymp 0 C

119894asymp 0 where C

119894asymp 0 are the secondary constraints

that arise from the requirement of the preservation of theconstraints 120587

119894asymp 0 These constraints can be solved for 120587

119894

and 119899119894 Further the first class constraint 120587119873asymp 0 can be gauge

fixed that leads to the elimination of 120587119873and119873 as dynamical

variables Finally the constraints Σ119894 H0can be again gauge

fixed which leads to the elimination of the Stuckelberg fieldsand conjugate momenta As a result we are left with 12degrees of freedom coming from the gravity sector that canbe identified with 10 degrees of freedom corresponding to themassive graviton and two degrees of freedom correspondingto the scalar at least at the linearized approximationNote thatthis scalar mode cannot be eliminated due to the absence ofthe scalar constraint As a result the Boulware-Deser ghost isgenerally present Finally there are two phase space degreesof freedom 120590 and 119901

120590

21 Remark about the Paper [22] After the first version ofthis paper was published Mukohyama argued in his paper[22] that the extension of the quasidilaton theory is ghost-free His arguments is based on the existence of the additionalconstraints that he imposed on the scalar fields

1206010

= minus119890minus120590119872119901

120601119894

= 120575119894

120583119909120583

119894 = 1 2 3 (29)

He called this constraint as the gauge fixing constraint andthen he was able to derive the Hamiltonian in the form

4 Journal of Gravity

119867 = int1198893x119873C

0 where the specific form of C

0is given in

[22] According to this result he claimed that there exists anadditional constraint in the gauge fixed theory so that thisconstraint is responsible for the elimination of the Boulware-Deser ghost

In this section we reconsider (29) from different point ofview and argue that this is not a gauge fixing constraint at allWe do not impose the fixing of spatial diffeomorphism andconsider the following relation between 120601

0and 120590 (note that

we do not use the word ldquogauge fixing conditionrdquo)

120590 = minus119872119901ln1206010 (30)

that is equivalent to (29) Inserting this relation to thedefinition of 119891

120583] we obtain

119891120583] = 120578119886119887120597120583120601

119886

120597]120601119887

minus

120572120590

1198982

119892

1205971205831206010

120597]1206010

(31)

Then inserting (31) into the dRGTmassive gravity we obtain

119878 = 1198722

119901int1198894

119909radicminus119892[(4)

119877 + 21198982

(3 minus

1

(1206010)2

radic119892minus1 119891)

minus

120596

2(1206010)2119892120583]1205971205831206010

120597]1206010

]

(32)

Now we see that given action is manifestly diffeomorphisminvariant In other words while we reduce the number ofdegrees of freedom by imposing (30) the number of gaugesymmetries is the same However we mean that the fact thatwe have less degrees of freedom than the original theorywhilethe number of gauge symmetries is the same implies thatthese two theories have different physical content and shouldnot be considered as equivalent In other words condition(29) is not a gauge fixing condition

From given analysis it is clear that the constraint C0

that was identified in [22] corresponds to the Hamiltonianconstraint in the theory with fixed spatial diffeomorphismClearly this constraint should have vanishing Poisson bracketC0(x)C

0(y) which implies that it is the first class con-

straint Clearly there is no way how to generate the additionalconstraint by imposing the requirement of the preservationof the constraint C

0during the time evolution of the system

since the Hamiltonian is equal to 119867 = int1198893x119873C0 However

sinceC0is the first class constraint it can be gauge fixed and

hence we reduce the number of physical degrees of freedomby two But we should again stress that the theory representedby action (33) is not equivalent to action (1) that representsthe extension of the quasidilaton dRGT theory In summarywe mean that the arguments that were presented in [22] donot prove that the extension of quasidilaton massive gravityis ghost-free

Let us compare this situation with the imposing staticgauge for the Hamiltonian (27) This gauge fixing is repre-sented by four gauge fixed functions

G0= 1206010

minus 119905 asymp 0 G119894= 120601119894

minus 119909119894

asymp 0 (33)

These constraints together withH0 Σ119894form the second class

constraints that can be explicitly solved for 119901119886 We solve H

0

for 1199010since we can identify the gauge fixed Hamiltonian

with 1199010= minusHgf The resulting Hamiltonian describes the

dynamic of the physical degrees of freedom 119892119894119895 120587119894119895 and 120590

119901120590 The detailed counting of the physical degrees of freedom

was presented in the end of previous section and we will notrepeat it here

3 The Case 120572120590= 0

In previous section we saw that the extension of the qua-sidilaton theory that was suggested in [14] suffers fromthe presence of Boulware-Deser ghost due to the absenceof two additional scalar constraints in the Hamiltonianformulation of given theory It is instructive to see whetherthese constraints emerge in the case of quasidilaton massivegravity which corresponds to the situation when 120572

120590= 0

Explicitly we consider the action [11 12]

119878 = 1198722

119901int1198893x119889119905 [119873radic119892

119894119895G119894119895119896119897

119896119897+ 119873radic119892119877 minus radic119892119872

0119880

minus 21198982

(119873radic119892radic119909Ω119863

119894

119894minus 3119873radic119892)

+ 119873radic119892

120596

1198722

119901

(nabla119899120590)2

minus 119873radic119892

120596

1198722

119901

120597119894120590119892119894119895

120597119895120590]

(34)

Let us now perform the Hamiltonian analysis of action (34)First of all we find the canonical momenta

119901119886= minus(minusM

119886119887

1

1198720

120597119905120601119887

119899119894

+ 119891119894119895

120597119895120601119886)R119894

+

1

1198720

1198722

119901radic119892M

119886119887120597119905120601119887

119880

+

1

1198720

M119886119887120597119905120601119887

119899119894

120597119894120590119901120590minus 120597119894120590119891119894119896

120597119896120601119886119901120590

119901120590= 120596radic119892nabla

119899120590

(35)

so that it is easy to find the scalar primary constraint

Σ119901equiv (119901119886+ (R119894+ 120597119894120590119901120590) 119891119894119896

120597119896120601119886) 120578119886119887

times (119901119887+ (R119894+ 120597119894120590119901120590) 119891119894119896

120597119896120601119887)

+ (119899119894

(R119894+ 119901120590120597119894120590) +119872

2

119901radic119892119880)

2

asymp 0

(36)

and also using the fact that 120597119894120601119886M119886119887= 0 we find additional

three constraints

Σ119894= 119901119886120597119894120601119886

+R119894+ 120597119894120590119901120590 (37)

Now using these constraints we can simplify (36) so that ithas the form

Σ119901= 119901119886M119886119887

119901119887+ (119899119894

(R119894+ 119901120590120597119894120590) +119872

2

119901radic119892119880)

2

asymp 0 (38)

Journal of Gravity 5

This has exactly the same form as the scalar constraint thatemerges in case of dRGT massive gravity written in theStuckelberg formalismThe minimal form of this gravity wasanalyzed in [16] and this analysis can be easily applied to ourcase From (15) we find the Hamiltonian with all primaryconstraints included

119867119864= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (39)

where we introduced the constraint

Σ119894= Σ119894+ 120597119894119899119895

120587119895+ 120597119895(119899119895

120587119894) (40)

and where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877 + 2119898

2

1198722

119901radic119892radic119909Ω119863

119894

119894

minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

(R119894+ 119901120590120597119894120590)

+

1

radic119892120596

1199012

120590+ 120596radic119892119892

119894119895

120597119894120590120597119895120590

(41)

As the next step we have to perform the analysis of thestability of the primary constraints120587

119894asymp 0120587

119873asymp 0 andΣ

119901asymp 0

In case of the constraint 120587119894asymp 0 we find

120597119905120587119894= 120587119894 119867 = minus[(R

119895+ 119901120590120597119895120590) minus

21198982

1198722

119901Ωradic119892

radic119909

119899119896

119891119896119895]

times [119873

120575 (119863119895

119898119899119898

)

120575119899119894

+ 120575119895

119894(119899119894

(R119894+ 119901120590120597119894120590)

+1198722

119901radic119892119880)] = 0

(42)

so that we impose the following secondary constraint

C119894=R119894+ 119901120590120597119894120590 minus

21198982

1198722

119901Ωradic119892

radic119909

119891119894119895119899119895

(43)

Now with the help of this constraint and the constraint Σ119894 we

can simplify Σ119901in a similar way as in [16]

Σ119901= 41198984

1198724

119901119892Ω2

+ 119901119860120578119860119861

119901119861 (44)

Then it is easy to show that Σ119901(x) Σ119901(y) = 0 and the

requirement of the preservation of given constraint leads tothe emergence of an additional constraint These constraintsare responsible for the elimination of Boulware-Deser ghostsee again [16] formore details In other words the presence ofthe kinetic term for the quasidilaton that minimally couplesto gravity does not spoil the property that the given theory isghost-free

4 Ghost-Free Extension of QuasidilatonMassive Gravity

We argued in Section 2 that the extension of quasidilatontheory as was formulated in [14] is plagued by the presenceof the Boulware-Deser ghost On the other hand giventheory has many nice properties so that it is desirable topropose its ghost-free version In this section we proposesuch a formulation when we replace the kinetic term for 120590by following tadpole galileon term

119878120590=minus119879int119889

4

119909Ψ (Φ119860

)radicminus det 119891120583] = minus119879int119889

4

119909Ψ (Φ119860

)119872radic119891

(45)

where Φ119860 = (120601119886 120590) 119891 = det 119891119894119895and where the function

Ψ(Φ119860

) was chosen in such a way that action (45) is invariantunder (4) We claim that the quasidilaton theory formulatedas the dRGT massive theory with 119891

120583] and with the kineticterm for the galileon given by (45) is ghost-free

To see this explicitly it is useful to introduce the followingnotation Let s write 119891

120583] as

119891120583] = 120597120583Φ

119860

G119860119861120597]Φ119861

(46)

where we introduced the metricG119860119861

G119860119861= (

120578119860119861

0

0 minus

120572120590

1198722

1199011198982119890minus2120590119872119901) (47)

We see that our proposal has the form of the galileon coupledto dRGT massive gravity [21] whose Hamiltonian analysiswas performed in [19] On the other hand the action definedby (45) is more complicated since the metric 120578

119860119861is replaced

with the more general metric G119860119861

that depends on Φ119860 andthere are also additional scalar fieldsΩ(120601119860)Ψ(Φ119860) Howeverwe can expect that this fact will not modify the constraintstructure of given theory

To see this explicitly let us briefly review theHamiltoniananalysis of the nonlinear massive gravity with the term (45)keeping in mind that more detailed analysis can be found in[19] As usual the momenta conjugate to119873 119899119894 and 119892

119894119895are

120587119873asymp 0 120587

119894asymp 0 120587

119894119895

= 1198722

119901radic119892G119894119895119896119897

119896119897

(48)

while the momentum conjugate to Φ119860 has the form

119901119860= minus(

120575119872

120575120597119905Φ119860119899119894

+G119860119861

119891119894119895

120597119895Φ119861

)R119894minus1198722

119901radic119892

120575119872

120597119905Φ1198601198801015840

(49)

where

1198801015840

= 21198982

Ω (Φ)radic119909 +

119879

1198722

119901

Ψ (Φ)radic119891 (50)

and where1198722 has the form

1198722

= minus120597119905Φ119860

M119860119861120597119905Φ119861

M119860119861= G119860119861minusG119860119862120597119894Φ119862 119891119894119895

120597119895Φ119863

G119863119861

(51)

6 Journal of Gravity

Note that the matrixM119860119861

obeys following relations

M119860119861G119861119862

M119862119863=M119860119863 120597

119894Φ119860

M119860119861= 0 (52)

Then it is easy to determine the following primary con-straints

Σ119901= (119899119894

R119894+1198722

119901radic1198921198801015840

)

2

+ (119901119860+R119894

119891119894119895

G119860119862120597119895Φ119862

)G119860119861

times (119901119861+R119894

119891119894119895

G119861119863120597119895Φ119863

) asymp 0

120597119894Φ119860

119901119860+R119894= Σ119894asymp 0

(53)

Now we are ready to write the extended Hamiltonian whichincludes all the primary constraints

119867119864= int1198893x (119873C

0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (54)

where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877

+ 21198982

1198722

119901radic119892Ω (Φ)

radic119909119863119894

119894minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

R119894

(55)

and where we introduced the constraints Σ119894defined as

Σ119894= Σ119894+ 120597119894119899119894

120587119894+ 120597119895(119899119895

120587119894) (56)

To proceed further we have to check the stability ofall constraints The procedure is the same as in [19] sothat we find that Σ

119894are the first class constraints while the

requirement of the preservation of the constraints 120587119894asymp 0

implies following secondary constraints [23 24]

C119894equivR119894minus

21198982

1198722

119901Ω (Φ)radic119892

radic119909

119891119894119895119899119895

asymp 0 (57)

Further the requirement of the preservation of the constraint120587119873asymp 0 implies an existence of the secondary constraintC

0asymp

0 Using the constraints C119894and Σ

119894 we replace the constraint

Σ119901by new independent constraint Σ

119901

Σ119901= 41198984

1198724

119901Ω2

119892 + 119901119860G119860119861

119901119861+ 2119879Ψradic

119891

times radic119901119860120597119894120601119860 119891119894119895120597119895120601119861119901119861+ 411989841198724

119901Ω2119892 + 1198792

Ψ2 119891 = 0

(58)

Then the total Hamiltonian with all constraints included hasthe form

119867119879= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894+ Γ119894

C119894)

(59)

Now we are ready to analyze the stability of all constraintsthat appear in (59) Again the analysis is the same as in [19]

with slight complication that now there are additional termsΨ(Φ) Ω(Φ) together with G

119860119861(Φ119860

) in the definition ofthe action However these terms are local functions of Φ119860so that they do not affect the result that Σ

119901(x) Σ119901(y) asymp

0 As a result the requirement of the preservation of theconstraint Σ

119901asymp 0 implies new constraint Σ119868119868

119901asymp 0 These two

constraints are the second class constraints that can be usedfor the elimination of the Boulware-Deser ghost mode and itsconjugate momenta

In this section we proposed an extension of the quasidila-ton massive gravity that is ghost-free This proposal can begeneralized in differentways either consider themost generalpotential of the dRGT massive gravity or more complicatedkinetic term for 120590 It would be also very interesting to analyzethe cosmological consequences of the model with action (45)following [14]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Grant agency of the Czechrepublic under the Grant P20112G028

References

[1] C de Rham and G Gabadadze ldquoGeneralization of the Fierz-Pauli actionrdquo Physical Review D vol 82 Article ID 0440202010

[2] C de Rham G Gabadadze and A J Tolley ldquoResummationof massive gravityrdquo Physical Review Letters vol 106 Article ID231101 2011

[3] D G Boulware and S Deser ldquoCan gravitation have a finiterangerdquo Physical Review D vol 6 no 12 pp 3368ndash3382 1972

[4] D G Boulware and S Desser ldquoInconsistency of finite rangegravitationrdquo Physics Letters B vol 40 no 2 pp 227ndash229 1972

[5] A de Felice A E Gumrukcuoglu and SMukohyama ldquoMassivegravity non-linear instability of the homogeneous and isotropicuniverserdquo Physical Review Letters vol 109 Article ID 1711012012

[6] K Koyama G Niz and G Tasinato ldquoThe self-acceleratinguniverse with vectors inmassive gravityrdquo Journal of High EnergyPhysics vol 2011 no 12 article 65 2011

[7] G Tasinato K Koyama and G Niz ldquoVector inst abilities andself-acceleration in the decoupling limit of massive gravityrdquoPhysical Review D vol 87 Article ID 064029 2013

[8] N Khosravi G Niz K Koyama and G Tasinato ldquoStabilityof the self-accelerating universe in massive gravityrdquo Journal ofCosmology and Astroparticle Physics vol 2013 no 8 p 44 2013

[9] G DrsquoAmico C de Rham S Dubovsky G Gabadadze DPirtskhalava and A J Tolley ldquoMassive cosmologiesrdquo PhysicalReview D vol 84 Article ID 124046 2011

[10] A E Gumrukcuoglu C Lin and S Mukohyama ldquoAnisotropicFriedmann-Robertson-Walker universe from nonlinear mas-sive gravityrdquoPhysics Letters B vol 717 no 4-5 pp 295ndash298 2012

Journal of Gravity 7

[11] A de Felice A E Gumrukcuoglu C Lin and S MukohyamaldquoNonlinear stability of cosmological solutions in massive grav-ityrdquo Journal of Cosmology andAstroparticle Physics vol 1305 no5 article 035 2013

[12] G DrsquoAmico G Gabadadze L Hui andD Pirtskhalava ldquoQuasi-dilaton theory and cosmologyrdquo Physical Review D vol 87Article ID 064037 2013

[13] Q-G Huang Y-S Piao and S-Y Zhou ldquoMass-varyingmassivegravityrdquo Physical Review D vol 86 Article ID 124014 2012

[14] A de Felice and S Mukohyama ldquoTowards consistent extensionof quasidilaton massive gravityrdquo Physics Letters B vol 728 pp622ndash625 2014

[15] G Gabadadze K Hinterbichler J Khoury D Pirtskhalavaand M Trodden ldquoCovariant master theory for novel Galileaninvariant models and massive gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 86 no 12Article ID 124004 2012

[16] J Kluson ldquoNote about Hamiltonian formalism for generalnonlinear massive gravity action in Stuckelberg formalismrdquoInternational Journal of Modern Physics A Particles and FieldsGravitation Cosmology vol 28 Article ID 1350160 16 pages2013

[17] J Kluson ldquoNon-linear massive gravity with additional primaryconstraint and absence of ghostsrdquo Physical Review D vol 86Article ID 044024 2012

[18] S F Hassan A Schmidt-May and M von Strauss ldquoProof ofconsistency of nonlinear massive gravity in the Stuckelbergformulationrdquo Physics Letters B vol 715 pp 355ndash339 2012

[19] J Kluson ldquoHamiltonian analysis of minimal massive gravitycoupled to Galileon tadpole termrdquo Journal of High EnergyPhysics vol 2013 article 80 2013

[20] Q G Huang K C Zhang and S Y Zhou ldquoGeneralizedmassive gravity in arbitrary dimensions and its Hamiltonianformulationrdquo Journal of Cosmology and Astroparticle Physicsvol 1308 article 050 2013

[21] M Andrews G Goon K Hinterbichler J Stokes and MTrodden ldquoMassive gravity coupled to Galileons is ghost-freerdquoPhysical Review Letters vol 111 no 6 Article ID 061107 2013

[22] SMukohyama ldquoExtended quasidilatonmassive gravity is ghostfreerdquo httparxivorgabs13092146

[23] S FHassan andRA Rosen ldquoOnnon-linear actions formassivegravityrdquo Journal of High Energy Physics vol 2011 article 9 2011

[24] S F Hassan and R A Rosen ldquoResolving the ghost problemin non-linear massive gravityrdquo Physical Review Letters vol 108Article ID 041101 2012

[25] E Gourgoulhon ldquo3 + 1 formalism and bases of numericalrelativityrdquo httparxivorgabsgr-qc0703035

[26] R L Arnowitt S Deser and C W Misner ldquoThe dynamics ofgeneral relativityrdquo httparxivorgabsgrqc0405109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in Condensed Matter Physics

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AstronomyAdvances in

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Superconductivity

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Statistical MechanicsInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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 Computational  Methods in Physics

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Soft MatterJournal of

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ThermodynamicsJournal of

4 Journal of Gravity

119867 = int1198893x119873C

0 where the specific form of C

0is given in

[22] According to this result he claimed that there exists anadditional constraint in the gauge fixed theory so that thisconstraint is responsible for the elimination of the Boulware-Deser ghost

In this section we reconsider (29) from different point ofview and argue that this is not a gauge fixing constraint at allWe do not impose the fixing of spatial diffeomorphism andconsider the following relation between 120601

0and 120590 (note that

we do not use the word ldquogauge fixing conditionrdquo)

120590 = minus119872119901ln1206010 (30)

that is equivalent to (29) Inserting this relation to thedefinition of 119891

120583] we obtain

119891120583] = 120578119886119887120597120583120601

119886

120597]120601119887

minus

120572120590

1198982

119892

1205971205831206010

120597]1206010

(31)

Then inserting (31) into the dRGTmassive gravity we obtain

119878 = 1198722

119901int1198894

119909radicminus119892[(4)

119877 + 21198982

(3 minus

1

(1206010)2

radic119892minus1 119891)

minus

120596

2(1206010)2119892120583]1205971205831206010

120597]1206010

]

(32)

Now we see that given action is manifestly diffeomorphisminvariant In other words while we reduce the number ofdegrees of freedom by imposing (30) the number of gaugesymmetries is the same However we mean that the fact thatwe have less degrees of freedom than the original theorywhilethe number of gauge symmetries is the same implies thatthese two theories have different physical content and shouldnot be considered as equivalent In other words condition(29) is not a gauge fixing condition

From given analysis it is clear that the constraint C0

that was identified in [22] corresponds to the Hamiltonianconstraint in the theory with fixed spatial diffeomorphismClearly this constraint should have vanishing Poisson bracketC0(x)C

0(y) which implies that it is the first class con-

straint Clearly there is no way how to generate the additionalconstraint by imposing the requirement of the preservationof the constraint C

0during the time evolution of the system

since the Hamiltonian is equal to 119867 = int1198893x119873C0 However

sinceC0is the first class constraint it can be gauge fixed and

hence we reduce the number of physical degrees of freedomby two But we should again stress that the theory representedby action (33) is not equivalent to action (1) that representsthe extension of the quasidilaton dRGT theory In summarywe mean that the arguments that were presented in [22] donot prove that the extension of quasidilaton massive gravityis ghost-free

Let us compare this situation with the imposing staticgauge for the Hamiltonian (27) This gauge fixing is repre-sented by four gauge fixed functions

G0= 1206010

minus 119905 asymp 0 G119894= 120601119894

minus 119909119894

asymp 0 (33)

These constraints together withH0 Σ119894form the second class

constraints that can be explicitly solved for 119901119886 We solve H

0

for 1199010since we can identify the gauge fixed Hamiltonian

with 1199010= minusHgf The resulting Hamiltonian describes the

dynamic of the physical degrees of freedom 119892119894119895 120587119894119895 and 120590

119901120590 The detailed counting of the physical degrees of freedom

was presented in the end of previous section and we will notrepeat it here

3 The Case 120572120590= 0

In previous section we saw that the extension of the qua-sidilaton theory that was suggested in [14] suffers fromthe presence of Boulware-Deser ghost due to the absenceof two additional scalar constraints in the Hamiltonianformulation of given theory It is instructive to see whetherthese constraints emerge in the case of quasidilaton massivegravity which corresponds to the situation when 120572

120590= 0

Explicitly we consider the action [11 12]

119878 = 1198722

119901int1198893x119889119905 [119873radic119892

119894119895G119894119895119896119897

119896119897+ 119873radic119892119877 minus radic119892119872

0119880

minus 21198982

(119873radic119892radic119909Ω119863

119894

119894minus 3119873radic119892)

+ 119873radic119892

120596

1198722

119901

(nabla119899120590)2

minus 119873radic119892

120596

1198722

119901

120597119894120590119892119894119895

120597119895120590]

(34)

Let us now perform the Hamiltonian analysis of action (34)First of all we find the canonical momenta

119901119886= minus(minusM

119886119887

1

1198720

120597119905120601119887

119899119894

+ 119891119894119895

120597119895120601119886)R119894

+

1

1198720

1198722

119901radic119892M

119886119887120597119905120601119887

119880

+

1

1198720

M119886119887120597119905120601119887

119899119894

120597119894120590119901120590minus 120597119894120590119891119894119896

120597119896120601119886119901120590

119901120590= 120596radic119892nabla

119899120590

(35)

so that it is easy to find the scalar primary constraint

Σ119901equiv (119901119886+ (R119894+ 120597119894120590119901120590) 119891119894119896

120597119896120601119886) 120578119886119887

times (119901119887+ (R119894+ 120597119894120590119901120590) 119891119894119896

120597119896120601119887)

+ (119899119894

(R119894+ 119901120590120597119894120590) +119872

2

119901radic119892119880)

2

asymp 0

(36)

and also using the fact that 120597119894120601119886M119886119887= 0 we find additional

three constraints

Σ119894= 119901119886120597119894120601119886

+R119894+ 120597119894120590119901120590 (37)

Now using these constraints we can simplify (36) so that ithas the form

Σ119901= 119901119886M119886119887

119901119887+ (119899119894

(R119894+ 119901120590120597119894120590) +119872

2

119901radic119892119880)

2

asymp 0 (38)

Journal of Gravity 5

This has exactly the same form as the scalar constraint thatemerges in case of dRGT massive gravity written in theStuckelberg formalismThe minimal form of this gravity wasanalyzed in [16] and this analysis can be easily applied to ourcase From (15) we find the Hamiltonian with all primaryconstraints included

119867119864= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (39)

where we introduced the constraint

Σ119894= Σ119894+ 120597119894119899119895

120587119895+ 120597119895(119899119895

120587119894) (40)

and where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877 + 2119898

2

1198722

119901radic119892radic119909Ω119863

119894

119894

minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

(R119894+ 119901120590120597119894120590)

+

1

radic119892120596

1199012

120590+ 120596radic119892119892

119894119895

120597119894120590120597119895120590

(41)

As the next step we have to perform the analysis of thestability of the primary constraints120587

119894asymp 0120587

119873asymp 0 andΣ

119901asymp 0

In case of the constraint 120587119894asymp 0 we find

120597119905120587119894= 120587119894 119867 = minus[(R

119895+ 119901120590120597119895120590) minus

21198982

1198722

119901Ωradic119892

radic119909

119899119896

119891119896119895]

times [119873

120575 (119863119895

119898119899119898

)

120575119899119894

+ 120575119895

119894(119899119894

(R119894+ 119901120590120597119894120590)

+1198722

119901radic119892119880)] = 0

(42)

so that we impose the following secondary constraint

C119894=R119894+ 119901120590120597119894120590 minus

21198982

1198722

119901Ωradic119892

radic119909

119891119894119895119899119895

(43)

Now with the help of this constraint and the constraint Σ119894 we

can simplify Σ119901in a similar way as in [16]

Σ119901= 41198984

1198724

119901119892Ω2

+ 119901119860120578119860119861

119901119861 (44)

Then it is easy to show that Σ119901(x) Σ119901(y) = 0 and the

requirement of the preservation of given constraint leads tothe emergence of an additional constraint These constraintsare responsible for the elimination of Boulware-Deser ghostsee again [16] formore details In other words the presence ofthe kinetic term for the quasidilaton that minimally couplesto gravity does not spoil the property that the given theory isghost-free

4 Ghost-Free Extension of QuasidilatonMassive Gravity

We argued in Section 2 that the extension of quasidilatontheory as was formulated in [14] is plagued by the presenceof the Boulware-Deser ghost On the other hand giventheory has many nice properties so that it is desirable topropose its ghost-free version In this section we proposesuch a formulation when we replace the kinetic term for 120590by following tadpole galileon term

119878120590=minus119879int119889

4

119909Ψ (Φ119860

)radicminus det 119891120583] = minus119879int119889

4

119909Ψ (Φ119860

)119872radic119891

(45)

where Φ119860 = (120601119886 120590) 119891 = det 119891119894119895and where the function

Ψ(Φ119860

) was chosen in such a way that action (45) is invariantunder (4) We claim that the quasidilaton theory formulatedas the dRGT massive theory with 119891

120583] and with the kineticterm for the galileon given by (45) is ghost-free

To see this explicitly it is useful to introduce the followingnotation Let s write 119891

120583] as

119891120583] = 120597120583Φ

119860

G119860119861120597]Φ119861

(46)

where we introduced the metricG119860119861

G119860119861= (

120578119860119861

0

0 minus

120572120590

1198722

1199011198982119890minus2120590119872119901) (47)

We see that our proposal has the form of the galileon coupledto dRGT massive gravity [21] whose Hamiltonian analysiswas performed in [19] On the other hand the action definedby (45) is more complicated since the metric 120578

119860119861is replaced

with the more general metric G119860119861

that depends on Φ119860 andthere are also additional scalar fieldsΩ(120601119860)Ψ(Φ119860) Howeverwe can expect that this fact will not modify the constraintstructure of given theory

To see this explicitly let us briefly review theHamiltoniananalysis of the nonlinear massive gravity with the term (45)keeping in mind that more detailed analysis can be found in[19] As usual the momenta conjugate to119873 119899119894 and 119892

119894119895are

120587119873asymp 0 120587

119894asymp 0 120587

119894119895

= 1198722

119901radic119892G119894119895119896119897

119896119897

(48)

while the momentum conjugate to Φ119860 has the form

119901119860= minus(

120575119872

120575120597119905Φ119860119899119894

+G119860119861

119891119894119895

120597119895Φ119861

)R119894minus1198722

119901radic119892

120575119872

120597119905Φ1198601198801015840

(49)

where

1198801015840

= 21198982

Ω (Φ)radic119909 +

119879

1198722

119901

Ψ (Φ)radic119891 (50)

and where1198722 has the form

1198722

= minus120597119905Φ119860

M119860119861120597119905Φ119861

M119860119861= G119860119861minusG119860119862120597119894Φ119862 119891119894119895

120597119895Φ119863

G119863119861

(51)

6 Journal of Gravity

Note that the matrixM119860119861

obeys following relations

M119860119861G119861119862

M119862119863=M119860119863 120597

119894Φ119860

M119860119861= 0 (52)

Then it is easy to determine the following primary con-straints

Σ119901= (119899119894

R119894+1198722

119901radic1198921198801015840

)

2

+ (119901119860+R119894

119891119894119895

G119860119862120597119895Φ119862

)G119860119861

times (119901119861+R119894

119891119894119895

G119861119863120597119895Φ119863

) asymp 0

120597119894Φ119860

119901119860+R119894= Σ119894asymp 0

(53)

Now we are ready to write the extended Hamiltonian whichincludes all the primary constraints

119867119864= int1198893x (119873C

0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (54)

where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877

+ 21198982

1198722

119901radic119892Ω (Φ)

radic119909119863119894

119894minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

R119894

(55)

and where we introduced the constraints Σ119894defined as

Σ119894= Σ119894+ 120597119894119899119894

120587119894+ 120597119895(119899119895

120587119894) (56)

To proceed further we have to check the stability ofall constraints The procedure is the same as in [19] sothat we find that Σ

119894are the first class constraints while the

requirement of the preservation of the constraints 120587119894asymp 0

implies following secondary constraints [23 24]

C119894equivR119894minus

21198982

1198722

119901Ω (Φ)radic119892

radic119909

119891119894119895119899119895

asymp 0 (57)

Further the requirement of the preservation of the constraint120587119873asymp 0 implies an existence of the secondary constraintC

0asymp

0 Using the constraints C119894and Σ

119894 we replace the constraint

Σ119901by new independent constraint Σ

119901

Σ119901= 41198984

1198724

119901Ω2

119892 + 119901119860G119860119861

119901119861+ 2119879Ψradic

119891

times radic119901119860120597119894120601119860 119891119894119895120597119895120601119861119901119861+ 411989841198724

119901Ω2119892 + 1198792

Ψ2 119891 = 0

(58)

Then the total Hamiltonian with all constraints included hasthe form

119867119879= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894+ Γ119894

C119894)

(59)

Now we are ready to analyze the stability of all constraintsthat appear in (59) Again the analysis is the same as in [19]

with slight complication that now there are additional termsΨ(Φ) Ω(Φ) together with G

119860119861(Φ119860

) in the definition ofthe action However these terms are local functions of Φ119860so that they do not affect the result that Σ

119901(x) Σ119901(y) asymp

0 As a result the requirement of the preservation of theconstraint Σ

119901asymp 0 implies new constraint Σ119868119868

119901asymp 0 These two

constraints are the second class constraints that can be usedfor the elimination of the Boulware-Deser ghost mode and itsconjugate momenta

In this section we proposed an extension of the quasidila-ton massive gravity that is ghost-free This proposal can begeneralized in differentways either consider themost generalpotential of the dRGT massive gravity or more complicatedkinetic term for 120590 It would be also very interesting to analyzethe cosmological consequences of the model with action (45)following [14]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Grant agency of the Czechrepublic under the Grant P20112G028

References

[1] C de Rham and G Gabadadze ldquoGeneralization of the Fierz-Pauli actionrdquo Physical Review D vol 82 Article ID 0440202010

[2] C de Rham G Gabadadze and A J Tolley ldquoResummationof massive gravityrdquo Physical Review Letters vol 106 Article ID231101 2011

[3] D G Boulware and S Deser ldquoCan gravitation have a finiterangerdquo Physical Review D vol 6 no 12 pp 3368ndash3382 1972

[4] D G Boulware and S Desser ldquoInconsistency of finite rangegravitationrdquo Physics Letters B vol 40 no 2 pp 227ndash229 1972

[5] A de Felice A E Gumrukcuoglu and SMukohyama ldquoMassivegravity non-linear instability of the homogeneous and isotropicuniverserdquo Physical Review Letters vol 109 Article ID 1711012012

[6] K Koyama G Niz and G Tasinato ldquoThe self-acceleratinguniverse with vectors inmassive gravityrdquo Journal of High EnergyPhysics vol 2011 no 12 article 65 2011

[7] G Tasinato K Koyama and G Niz ldquoVector inst abilities andself-acceleration in the decoupling limit of massive gravityrdquoPhysical Review D vol 87 Article ID 064029 2013

[8] N Khosravi G Niz K Koyama and G Tasinato ldquoStabilityof the self-accelerating universe in massive gravityrdquo Journal ofCosmology and Astroparticle Physics vol 2013 no 8 p 44 2013

[9] G DrsquoAmico C de Rham S Dubovsky G Gabadadze DPirtskhalava and A J Tolley ldquoMassive cosmologiesrdquo PhysicalReview D vol 84 Article ID 124046 2011

[10] A E Gumrukcuoglu C Lin and S Mukohyama ldquoAnisotropicFriedmann-Robertson-Walker universe from nonlinear mas-sive gravityrdquoPhysics Letters B vol 717 no 4-5 pp 295ndash298 2012

Journal of Gravity 7

[11] A de Felice A E Gumrukcuoglu C Lin and S MukohyamaldquoNonlinear stability of cosmological solutions in massive grav-ityrdquo Journal of Cosmology andAstroparticle Physics vol 1305 no5 article 035 2013

[12] G DrsquoAmico G Gabadadze L Hui andD Pirtskhalava ldquoQuasi-dilaton theory and cosmologyrdquo Physical Review D vol 87Article ID 064037 2013

[13] Q-G Huang Y-S Piao and S-Y Zhou ldquoMass-varyingmassivegravityrdquo Physical Review D vol 86 Article ID 124014 2012

[14] A de Felice and S Mukohyama ldquoTowards consistent extensionof quasidilaton massive gravityrdquo Physics Letters B vol 728 pp622ndash625 2014

[15] G Gabadadze K Hinterbichler J Khoury D Pirtskhalavaand M Trodden ldquoCovariant master theory for novel Galileaninvariant models and massive gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 86 no 12Article ID 124004 2012

[16] J Kluson ldquoNote about Hamiltonian formalism for generalnonlinear massive gravity action in Stuckelberg formalismrdquoInternational Journal of Modern Physics A Particles and FieldsGravitation Cosmology vol 28 Article ID 1350160 16 pages2013

[17] J Kluson ldquoNon-linear massive gravity with additional primaryconstraint and absence of ghostsrdquo Physical Review D vol 86Article ID 044024 2012

[18] S F Hassan A Schmidt-May and M von Strauss ldquoProof ofconsistency of nonlinear massive gravity in the Stuckelbergformulationrdquo Physics Letters B vol 715 pp 355ndash339 2012

[19] J Kluson ldquoHamiltonian analysis of minimal massive gravitycoupled to Galileon tadpole termrdquo Journal of High EnergyPhysics vol 2013 article 80 2013

[20] Q G Huang K C Zhang and S Y Zhou ldquoGeneralizedmassive gravity in arbitrary dimensions and its Hamiltonianformulationrdquo Journal of Cosmology and Astroparticle Physicsvol 1308 article 050 2013

[21] M Andrews G Goon K Hinterbichler J Stokes and MTrodden ldquoMassive gravity coupled to Galileons is ghost-freerdquoPhysical Review Letters vol 111 no 6 Article ID 061107 2013

[22] SMukohyama ldquoExtended quasidilatonmassive gravity is ghostfreerdquo httparxivorgabs13092146

[23] S FHassan andRA Rosen ldquoOnnon-linear actions formassivegravityrdquo Journal of High Energy Physics vol 2011 article 9 2011

[24] S F Hassan and R A Rosen ldquoResolving the ghost problemin non-linear massive gravityrdquo Physical Review Letters vol 108Article ID 041101 2012

[25] E Gourgoulhon ldquo3 + 1 formalism and bases of numericalrelativityrdquo httparxivorgabsgr-qc0703035

[26] R L Arnowitt S Deser and C W Misner ldquoThe dynamics ofgeneral relativityrdquo httparxivorgabsgrqc0405109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Gravity 5

This has exactly the same form as the scalar constraint thatemerges in case of dRGT massive gravity written in theStuckelberg formalismThe minimal form of this gravity wasanalyzed in [16] and this analysis can be easily applied to ourcase From (15) we find the Hamiltonian with all primaryconstraints included

119867119864= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (39)

where we introduced the constraint

Σ119894= Σ119894+ 120597119894119899119895

120587119895+ 120597119895(119899119895

120587119894) (40)

and where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877 + 2119898

2

1198722

119901radic119892radic119909Ω119863

119894

119894

minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

(R119894+ 119901120590120597119894120590)

+

1

radic119892120596

1199012

120590+ 120596radic119892119892

119894119895

120597119894120590120597119895120590

(41)

As the next step we have to perform the analysis of thestability of the primary constraints120587

119894asymp 0120587

119873asymp 0 andΣ

119901asymp 0

In case of the constraint 120587119894asymp 0 we find

120597119905120587119894= 120587119894 119867 = minus[(R

119895+ 119901120590120597119895120590) minus

21198982

1198722

119901Ωradic119892

radic119909

119899119896

119891119896119895]

times [119873

120575 (119863119895

119898119899119898

)

120575119899119894

+ 120575119895

119894(119899119894

(R119894+ 119901120590120597119894120590)

+1198722

119901radic119892119880)] = 0

(42)

so that we impose the following secondary constraint

C119894=R119894+ 119901120590120597119894120590 minus

21198982

1198722

119901Ωradic119892

radic119909

119891119894119895119899119895

(43)

Now with the help of this constraint and the constraint Σ119894 we

can simplify Σ119901in a similar way as in [16]

Σ119901= 41198984

1198724

119901119892Ω2

+ 119901119860120578119860119861

119901119861 (44)

Then it is easy to show that Σ119901(x) Σ119901(y) = 0 and the

requirement of the preservation of given constraint leads tothe emergence of an additional constraint These constraintsare responsible for the elimination of Boulware-Deser ghostsee again [16] formore details In other words the presence ofthe kinetic term for the quasidilaton that minimally couplesto gravity does not spoil the property that the given theory isghost-free

4 Ghost-Free Extension of QuasidilatonMassive Gravity

We argued in Section 2 that the extension of quasidilatontheory as was formulated in [14] is plagued by the presenceof the Boulware-Deser ghost On the other hand giventheory has many nice properties so that it is desirable topropose its ghost-free version In this section we proposesuch a formulation when we replace the kinetic term for 120590by following tadpole galileon term

119878120590=minus119879int119889

4

119909Ψ (Φ119860

)radicminus det 119891120583] = minus119879int119889

4

119909Ψ (Φ119860

)119872radic119891

(45)

where Φ119860 = (120601119886 120590) 119891 = det 119891119894119895and where the function

Ψ(Φ119860

) was chosen in such a way that action (45) is invariantunder (4) We claim that the quasidilaton theory formulatedas the dRGT massive theory with 119891

120583] and with the kineticterm for the galileon given by (45) is ghost-free

To see this explicitly it is useful to introduce the followingnotation Let s write 119891

120583] as

119891120583] = 120597120583Φ

119860

G119860119861120597]Φ119861

(46)

where we introduced the metricG119860119861

G119860119861= (

120578119860119861

0

0 minus

120572120590

1198722

1199011198982119890minus2120590119872119901) (47)

We see that our proposal has the form of the galileon coupledto dRGT massive gravity [21] whose Hamiltonian analysiswas performed in [19] On the other hand the action definedby (45) is more complicated since the metric 120578

119860119861is replaced

with the more general metric G119860119861

that depends on Φ119860 andthere are also additional scalar fieldsΩ(120601119860)Ψ(Φ119860) Howeverwe can expect that this fact will not modify the constraintstructure of given theory

To see this explicitly let us briefly review theHamiltoniananalysis of the nonlinear massive gravity with the term (45)keeping in mind that more detailed analysis can be found in[19] As usual the momenta conjugate to119873 119899119894 and 119892

119894119895are

120587119873asymp 0 120587

119894asymp 0 120587

119894119895

= 1198722

119901radic119892G119894119895119896119897

119896119897

(48)

while the momentum conjugate to Φ119860 has the form

119901119860= minus(

120575119872

120575120597119905Φ119860119899119894

+G119860119861

119891119894119895

120597119895Φ119861

)R119894minus1198722

119901radic119892

120575119872

120597119905Φ1198601198801015840

(49)

where

1198801015840

= 21198982

Ω (Φ)radic119909 +

119879

1198722

119901

Ψ (Φ)radic119891 (50)

and where1198722 has the form

1198722

= minus120597119905Φ119860

M119860119861120597119905Φ119861

M119860119861= G119860119861minusG119860119862120597119894Φ119862 119891119894119895

120597119895Φ119863

G119863119861

(51)

6 Journal of Gravity

Note that the matrixM119860119861

obeys following relations

M119860119861G119861119862

M119862119863=M119860119863 120597

119894Φ119860

M119860119861= 0 (52)

Then it is easy to determine the following primary con-straints

Σ119901= (119899119894

R119894+1198722

119901radic1198921198801015840

)

2

+ (119901119860+R119894

119891119894119895

G119860119862120597119895Φ119862

)G119860119861

times (119901119861+R119894

119891119894119895

G119861119863120597119895Φ119863

) asymp 0

120597119894Φ119860

119901119860+R119894= Σ119894asymp 0

(53)

Now we are ready to write the extended Hamiltonian whichincludes all the primary constraints

119867119864= int1198893x (119873C

0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (54)

where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877

+ 21198982

1198722

119901radic119892Ω (Φ)

radic119909119863119894

119894minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

R119894

(55)

and where we introduced the constraints Σ119894defined as

Σ119894= Σ119894+ 120597119894119899119894

120587119894+ 120597119895(119899119895

120587119894) (56)

To proceed further we have to check the stability ofall constraints The procedure is the same as in [19] sothat we find that Σ

119894are the first class constraints while the

requirement of the preservation of the constraints 120587119894asymp 0

implies following secondary constraints [23 24]

C119894equivR119894minus

21198982

1198722

119901Ω (Φ)radic119892

radic119909

119891119894119895119899119895

asymp 0 (57)

Further the requirement of the preservation of the constraint120587119873asymp 0 implies an existence of the secondary constraintC

0asymp

0 Using the constraints C119894and Σ

119894 we replace the constraint

Σ119901by new independent constraint Σ

119901

Σ119901= 41198984

1198724

119901Ω2

119892 + 119901119860G119860119861

119901119861+ 2119879Ψradic

119891

times radic119901119860120597119894120601119860 119891119894119895120597119895120601119861119901119861+ 411989841198724

119901Ω2119892 + 1198792

Ψ2 119891 = 0

(58)

Then the total Hamiltonian with all constraints included hasthe form

119867119879= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894+ Γ119894

C119894)

(59)

Now we are ready to analyze the stability of all constraintsthat appear in (59) Again the analysis is the same as in [19]

with slight complication that now there are additional termsΨ(Φ) Ω(Φ) together with G

119860119861(Φ119860

) in the definition ofthe action However these terms are local functions of Φ119860so that they do not affect the result that Σ

119901(x) Σ119901(y) asymp

0 As a result the requirement of the preservation of theconstraint Σ

119901asymp 0 implies new constraint Σ119868119868

119901asymp 0 These two

constraints are the second class constraints that can be usedfor the elimination of the Boulware-Deser ghost mode and itsconjugate momenta

In this section we proposed an extension of the quasidila-ton massive gravity that is ghost-free This proposal can begeneralized in differentways either consider themost generalpotential of the dRGT massive gravity or more complicatedkinetic term for 120590 It would be also very interesting to analyzethe cosmological consequences of the model with action (45)following [14]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Grant agency of the Czechrepublic under the Grant P20112G028

References

[1] C de Rham and G Gabadadze ldquoGeneralization of the Fierz-Pauli actionrdquo Physical Review D vol 82 Article ID 0440202010

[2] C de Rham G Gabadadze and A J Tolley ldquoResummationof massive gravityrdquo Physical Review Letters vol 106 Article ID231101 2011

[3] D G Boulware and S Deser ldquoCan gravitation have a finiterangerdquo Physical Review D vol 6 no 12 pp 3368ndash3382 1972

[4] D G Boulware and S Desser ldquoInconsistency of finite rangegravitationrdquo Physics Letters B vol 40 no 2 pp 227ndash229 1972

[5] A de Felice A E Gumrukcuoglu and SMukohyama ldquoMassivegravity non-linear instability of the homogeneous and isotropicuniverserdquo Physical Review Letters vol 109 Article ID 1711012012

[6] K Koyama G Niz and G Tasinato ldquoThe self-acceleratinguniverse with vectors inmassive gravityrdquo Journal of High EnergyPhysics vol 2011 no 12 article 65 2011

[7] G Tasinato K Koyama and G Niz ldquoVector inst abilities andself-acceleration in the decoupling limit of massive gravityrdquoPhysical Review D vol 87 Article ID 064029 2013

[8] N Khosravi G Niz K Koyama and G Tasinato ldquoStabilityof the self-accelerating universe in massive gravityrdquo Journal ofCosmology and Astroparticle Physics vol 2013 no 8 p 44 2013

[9] G DrsquoAmico C de Rham S Dubovsky G Gabadadze DPirtskhalava and A J Tolley ldquoMassive cosmologiesrdquo PhysicalReview D vol 84 Article ID 124046 2011

[10] A E Gumrukcuoglu C Lin and S Mukohyama ldquoAnisotropicFriedmann-Robertson-Walker universe from nonlinear mas-sive gravityrdquoPhysics Letters B vol 717 no 4-5 pp 295ndash298 2012

Journal of Gravity 7

[11] A de Felice A E Gumrukcuoglu C Lin and S MukohyamaldquoNonlinear stability of cosmological solutions in massive grav-ityrdquo Journal of Cosmology andAstroparticle Physics vol 1305 no5 article 035 2013

[12] G DrsquoAmico G Gabadadze L Hui andD Pirtskhalava ldquoQuasi-dilaton theory and cosmologyrdquo Physical Review D vol 87Article ID 064037 2013

[13] Q-G Huang Y-S Piao and S-Y Zhou ldquoMass-varyingmassivegravityrdquo Physical Review D vol 86 Article ID 124014 2012

[14] A de Felice and S Mukohyama ldquoTowards consistent extensionof quasidilaton massive gravityrdquo Physics Letters B vol 728 pp622ndash625 2014

[15] G Gabadadze K Hinterbichler J Khoury D Pirtskhalavaand M Trodden ldquoCovariant master theory for novel Galileaninvariant models and massive gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 86 no 12Article ID 124004 2012

[16] J Kluson ldquoNote about Hamiltonian formalism for generalnonlinear massive gravity action in Stuckelberg formalismrdquoInternational Journal of Modern Physics A Particles and FieldsGravitation Cosmology vol 28 Article ID 1350160 16 pages2013

[17] J Kluson ldquoNon-linear massive gravity with additional primaryconstraint and absence of ghostsrdquo Physical Review D vol 86Article ID 044024 2012

[18] S F Hassan A Schmidt-May and M von Strauss ldquoProof ofconsistency of nonlinear massive gravity in the Stuckelbergformulationrdquo Physics Letters B vol 715 pp 355ndash339 2012

[19] J Kluson ldquoHamiltonian analysis of minimal massive gravitycoupled to Galileon tadpole termrdquo Journal of High EnergyPhysics vol 2013 article 80 2013

[20] Q G Huang K C Zhang and S Y Zhou ldquoGeneralizedmassive gravity in arbitrary dimensions and its Hamiltonianformulationrdquo Journal of Cosmology and Astroparticle Physicsvol 1308 article 050 2013

[21] M Andrews G Goon K Hinterbichler J Stokes and MTrodden ldquoMassive gravity coupled to Galileons is ghost-freerdquoPhysical Review Letters vol 111 no 6 Article ID 061107 2013

[22] SMukohyama ldquoExtended quasidilatonmassive gravity is ghostfreerdquo httparxivorgabs13092146

[23] S FHassan andRA Rosen ldquoOnnon-linear actions formassivegravityrdquo Journal of High Energy Physics vol 2011 article 9 2011

[24] S F Hassan and R A Rosen ldquoResolving the ghost problemin non-linear massive gravityrdquo Physical Review Letters vol 108Article ID 041101 2012

[25] E Gourgoulhon ldquo3 + 1 formalism and bases of numericalrelativityrdquo httparxivorgabsgr-qc0703035

[26] R L Arnowitt S Deser and C W Misner ldquoThe dynamics ofgeneral relativityrdquo httparxivorgabsgrqc0405109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Journal of Gravity

Note that the matrixM119860119861

obeys following relations

M119860119861G119861119862

M119862119863=M119860119863 120597

119894Φ119860

M119860119861= 0 (52)

Then it is easy to determine the following primary con-straints

Σ119901= (119899119894

R119894+1198722

119901radic1198921198801015840

)

2

+ (119901119860+R119894

119891119894119895

G119860119862120597119895Φ119862

)G119860119861

times (119901119861+R119894

119891119894119895

G119861119863120597119895Φ119863

) asymp 0

120597119894Φ119860

119901119860+R119894= Σ119894asymp 0

(53)

Now we are ready to write the extended Hamiltonian whichincludes all the primary constraints

119867119864= int1198893x (119873C

0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894) (54)

where

C0=

1

radic1198921198722

119901

120587119894119895

G119894119895119896119897120587119896119897

minus1198722

119901radic119892119877

+ 21198982

1198722

119901radic119892Ω (Φ)

radic119909119863119894

119894minus 61198982

1198722

119901radic119892 + 119863

119894

119895119899119895

R119894

(55)

and where we introduced the constraints Σ119894defined as

Σ119894= Σ119894+ 120597119894119899119894

120587119894+ 120597119895(119899119895

120587119894) (56)

To proceed further we have to check the stability ofall constraints The procedure is the same as in [19] sothat we find that Σ

119894are the first class constraints while the

requirement of the preservation of the constraints 120587119894asymp 0

implies following secondary constraints [23 24]

C119894equivR119894minus

21198982

1198722

119901Ω (Φ)radic119892

radic119909

119891119894119895119899119895

asymp 0 (57)

Further the requirement of the preservation of the constraint120587119873asymp 0 implies an existence of the secondary constraintC

0asymp

0 Using the constraints C119894and Σ

119894 we replace the constraint

Σ119901by new independent constraint Σ

119901

Σ119901= 41198984

1198724

119901Ω2

119892 + 119901119860G119860119861

119901119861+ 2119879Ψradic

119891

times radic119901119860120597119894120601119860 119891119894119895120597119895120601119861119901119861+ 411989841198724

119901Ω2119892 + 1198792

Ψ2 119891 = 0

(58)

Then the total Hamiltonian with all constraints included hasthe form

119867119879= int119889

3x (119873C0+ V119873120587119873+ V119894120587119894+ Ω119901Σ119901+ Ω119894

Σ119894+ Γ119894

C119894)

(59)

Now we are ready to analyze the stability of all constraintsthat appear in (59) Again the analysis is the same as in [19]

with slight complication that now there are additional termsΨ(Φ) Ω(Φ) together with G

119860119861(Φ119860

) in the definition ofthe action However these terms are local functions of Φ119860so that they do not affect the result that Σ

119901(x) Σ119901(y) asymp

0 As a result the requirement of the preservation of theconstraint Σ

119901asymp 0 implies new constraint Σ119868119868

119901asymp 0 These two

constraints are the second class constraints that can be usedfor the elimination of the Boulware-Deser ghost mode and itsconjugate momenta

In this section we proposed an extension of the quasidila-ton massive gravity that is ghost-free This proposal can begeneralized in differentways either consider themost generalpotential of the dRGT massive gravity or more complicatedkinetic term for 120590 It would be also very interesting to analyzethe cosmological consequences of the model with action (45)following [14]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Grant agency of the Czechrepublic under the Grant P20112G028

References

[1] C de Rham and G Gabadadze ldquoGeneralization of the Fierz-Pauli actionrdquo Physical Review D vol 82 Article ID 0440202010

[2] C de Rham G Gabadadze and A J Tolley ldquoResummationof massive gravityrdquo Physical Review Letters vol 106 Article ID231101 2011

[3] D G Boulware and S Deser ldquoCan gravitation have a finiterangerdquo Physical Review D vol 6 no 12 pp 3368ndash3382 1972

[4] D G Boulware and S Desser ldquoInconsistency of finite rangegravitationrdquo Physics Letters B vol 40 no 2 pp 227ndash229 1972

[5] A de Felice A E Gumrukcuoglu and SMukohyama ldquoMassivegravity non-linear instability of the homogeneous and isotropicuniverserdquo Physical Review Letters vol 109 Article ID 1711012012

[6] K Koyama G Niz and G Tasinato ldquoThe self-acceleratinguniverse with vectors inmassive gravityrdquo Journal of High EnergyPhysics vol 2011 no 12 article 65 2011

[7] G Tasinato K Koyama and G Niz ldquoVector inst abilities andself-acceleration in the decoupling limit of massive gravityrdquoPhysical Review D vol 87 Article ID 064029 2013

[8] N Khosravi G Niz K Koyama and G Tasinato ldquoStabilityof the self-accelerating universe in massive gravityrdquo Journal ofCosmology and Astroparticle Physics vol 2013 no 8 p 44 2013

[9] G DrsquoAmico C de Rham S Dubovsky G Gabadadze DPirtskhalava and A J Tolley ldquoMassive cosmologiesrdquo PhysicalReview D vol 84 Article ID 124046 2011

[10] A E Gumrukcuoglu C Lin and S Mukohyama ldquoAnisotropicFriedmann-Robertson-Walker universe from nonlinear mas-sive gravityrdquoPhysics Letters B vol 717 no 4-5 pp 295ndash298 2012

Journal of Gravity 7

[11] A de Felice A E Gumrukcuoglu C Lin and S MukohyamaldquoNonlinear stability of cosmological solutions in massive grav-ityrdquo Journal of Cosmology andAstroparticle Physics vol 1305 no5 article 035 2013

[12] G DrsquoAmico G Gabadadze L Hui andD Pirtskhalava ldquoQuasi-dilaton theory and cosmologyrdquo Physical Review D vol 87Article ID 064037 2013

[13] Q-G Huang Y-S Piao and S-Y Zhou ldquoMass-varyingmassivegravityrdquo Physical Review D vol 86 Article ID 124014 2012

[14] A de Felice and S Mukohyama ldquoTowards consistent extensionof quasidilaton massive gravityrdquo Physics Letters B vol 728 pp622ndash625 2014

[15] G Gabadadze K Hinterbichler J Khoury D Pirtskhalavaand M Trodden ldquoCovariant master theory for novel Galileaninvariant models and massive gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 86 no 12Article ID 124004 2012

[16] J Kluson ldquoNote about Hamiltonian formalism for generalnonlinear massive gravity action in Stuckelberg formalismrdquoInternational Journal of Modern Physics A Particles and FieldsGravitation Cosmology vol 28 Article ID 1350160 16 pages2013

[17] J Kluson ldquoNon-linear massive gravity with additional primaryconstraint and absence of ghostsrdquo Physical Review D vol 86Article ID 044024 2012

[18] S F Hassan A Schmidt-May and M von Strauss ldquoProof ofconsistency of nonlinear massive gravity in the Stuckelbergformulationrdquo Physics Letters B vol 715 pp 355ndash339 2012

[19] J Kluson ldquoHamiltonian analysis of minimal massive gravitycoupled to Galileon tadpole termrdquo Journal of High EnergyPhysics vol 2013 article 80 2013

[20] Q G Huang K C Zhang and S Y Zhou ldquoGeneralizedmassive gravity in arbitrary dimensions and its Hamiltonianformulationrdquo Journal of Cosmology and Astroparticle Physicsvol 1308 article 050 2013

[21] M Andrews G Goon K Hinterbichler J Stokes and MTrodden ldquoMassive gravity coupled to Galileons is ghost-freerdquoPhysical Review Letters vol 111 no 6 Article ID 061107 2013

[22] SMukohyama ldquoExtended quasidilatonmassive gravity is ghostfreerdquo httparxivorgabs13092146

[23] S FHassan andRA Rosen ldquoOnnon-linear actions formassivegravityrdquo Journal of High Energy Physics vol 2011 article 9 2011

[24] S F Hassan and R A Rosen ldquoResolving the ghost problemin non-linear massive gravityrdquo Physical Review Letters vol 108Article ID 041101 2012

[25] E Gourgoulhon ldquo3 + 1 formalism and bases of numericalrelativityrdquo httparxivorgabsgr-qc0703035

[26] R L Arnowitt S Deser and C W Misner ldquoThe dynamics ofgeneral relativityrdquo httparxivorgabsgrqc0405109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Gravity 7

[11] A de Felice A E Gumrukcuoglu C Lin and S MukohyamaldquoNonlinear stability of cosmological solutions in massive grav-ityrdquo Journal of Cosmology andAstroparticle Physics vol 1305 no5 article 035 2013

[12] G DrsquoAmico G Gabadadze L Hui andD Pirtskhalava ldquoQuasi-dilaton theory and cosmologyrdquo Physical Review D vol 87Article ID 064037 2013

[13] Q-G Huang Y-S Piao and S-Y Zhou ldquoMass-varyingmassivegravityrdquo Physical Review D vol 86 Article ID 124014 2012

[14] A de Felice and S Mukohyama ldquoTowards consistent extensionof quasidilaton massive gravityrdquo Physics Letters B vol 728 pp622ndash625 2014

[15] G Gabadadze K Hinterbichler J Khoury D Pirtskhalavaand M Trodden ldquoCovariant master theory for novel Galileaninvariant models and massive gravityrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 86 no 12Article ID 124004 2012

[16] J Kluson ldquoNote about Hamiltonian formalism for generalnonlinear massive gravity action in Stuckelberg formalismrdquoInternational Journal of Modern Physics A Particles and FieldsGravitation Cosmology vol 28 Article ID 1350160 16 pages2013

[17] J Kluson ldquoNon-linear massive gravity with additional primaryconstraint and absence of ghostsrdquo Physical Review D vol 86Article ID 044024 2012

[18] S F Hassan A Schmidt-May and M von Strauss ldquoProof ofconsistency of nonlinear massive gravity in the Stuckelbergformulationrdquo Physics Letters B vol 715 pp 355ndash339 2012

[19] J Kluson ldquoHamiltonian analysis of minimal massive gravitycoupled to Galileon tadpole termrdquo Journal of High EnergyPhysics vol 2013 article 80 2013

[20] Q G Huang K C Zhang and S Y Zhou ldquoGeneralizedmassive gravity in arbitrary dimensions and its Hamiltonianformulationrdquo Journal of Cosmology and Astroparticle Physicsvol 1308 article 050 2013

[21] M Andrews G Goon K Hinterbichler J Stokes and MTrodden ldquoMassive gravity coupled to Galileons is ghost-freerdquoPhysical Review Letters vol 111 no 6 Article ID 061107 2013

[22] SMukohyama ldquoExtended quasidilatonmassive gravity is ghostfreerdquo httparxivorgabs13092146

[23] S FHassan andRA Rosen ldquoOnnon-linear actions formassivegravityrdquo Journal of High Energy Physics vol 2011 article 9 2011

[24] S F Hassan and R A Rosen ldquoResolving the ghost problemin non-linear massive gravityrdquo Physical Review Letters vol 108Article ID 041101 2012

[25] E Gourgoulhon ldquo3 + 1 formalism and bases of numericalrelativityrdquo httparxivorgabsgr-qc0703035

[26] R L Arnowitt S Deser and C W Misner ldquoThe dynamics ofgeneral relativityrdquo httparxivorgabsgrqc0405109

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of