Chiral description of massive gravity Sergey Alexandrov, Kirill Krasnov, Simone Speziale

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Sergey Alexandrov, Kirill Krasnov, Simone Speziale. Chiral description of massive gravity. Journal of High Energy Physics, Springer, 2013, 2013 (6), pp.068. ￿10.1007/JHEP06(2013)068￿. ￿hal-00766632￿

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Distributed under a Creative Commons Attribution| 4.0 International License JHEP06(2013)068 Springer June 1, 2013 June 17, 2013 April 30, 2013 : : : January 10, 2013 e massive grav- : Revised Accepted Published 10.1007/JHEP06(2013)068 Received nd are left as an open issue. doi: ther with Plebanski’s chiral of freedom, consistently with t the stabilization procedure, d in the Lorentzian signature e structure similar to that of d s of the system. In particular, in terms of 2-forms opens the putations of the constraint alge- tions. Our results apply directly R 5221, Published for SISSA by , [email protected] , and Simone Speziale c Kirill Krasnov a,b 1212.3614 Classical Theories of Gravity, Models of Quantum Gravity We propose and study a new first order version of the ghost-fre [email protected] [email protected] E-mail: F-34095, Montpellier, France Mathematical Sciences, University ofNottingham, Nottingham, NG7 2RD, U.K. Centre de Physique CNRS-UMR Th´eorique, 7332, Luminy Case 907, 13288 Marseille, France Universit´eMontpellier 2, Laboratoire Charles CoulombF-34095, UM Montpellier, France CNRS, Laboratoire Charles Coulomb UMR 5221, b c d a Open Access ArXiv ePrint: door to an infinite classto of Euclidean ghost-free signature. massive bi-gravity The ac appear reality to conditions be more to complicated be than impose in the usual gravityKeywords: case a SU(2) gauge theories. The chiralbra, description and simplifies allows com us towe perform explicitly the compute complete the canonical secondary analysi thus proving constraint that and in carry general ou previous the claims. theory propagates Finally, 7 we degrees point out that the description Abstract: ity. Instead of2-forms metrics as or fundamental tetrads, variables, it rendering uses the phase a spac connection toge Sergei Alexandrov, Chiral description of massive gravity JHEP06(2013)068 ] 4 6 1 4 8 3 10 12 14 16 18 20 22 23 26 10 20 -linear completions of the , and a most recent wave of ically natural solution to the egree of freedom. Moreover, it infrared is to give the graviton egree of freedom with various e this, the ghost was argued [ ]. 6 ] suggested that a certain non-linear 9 ]. Perturbatively, the ghost manifests – 2 7 ] of alleviating the so-called van Dam-Veltman- 4 – 1 – = 0 ] and references therein. The theory is, however, 1 a ˜ H ]. A related difficulty with massive gravity is that the theory 5 Recently de Rham, Gabadadze and Tolley [ 4.6 Modified Plebanski theory 4.1 3 +4.2 1 decomposition and Primary field constraints 4.3 redefinition Secondary constraint 4.4 Stability condition4.5 for the secondary Summary constraint 2.1 Plebanski formulation2.2 of general relativity Canonical analysis to play a role in theZakharov Vainshtein mechanism [ discontinuity [ becomes strongly coupled at a very low energy scalescompletion [ of the Fierz-Pauli theory is free from the ghost d plagued by many difficulties.Fierz-Pauli One linear of theory them is propagatepathologies, that an the generical additional so non scalar calleditself as Boulware-Deser d a ghost scalar field [ with a wrong sign kinetic term. Despit interest comes from a possibilitycosmological constant that problem, it see may e.g. provide [ a techn 1 Introduction A deceptively simple way to modifya general small relativity mass. in the Massive gravity is a subject with a long history B Perturbative analysis around C Commutator of two Hamiltonian constraints 5 Conclusions A Evaluation of the secondary constraint 3 A chiral bi-gravity model 4 Canonical analysis and ghost-freeness Contents 1 Introduction 2 Summary of the chiral Plebanski formulation JHEP06(2013)068 ) and 1.2 (1.2) (1.1) ] and + . µν for the 14 g L − I − e e ∧ K − e and ∧ I + e J − ic theory of gravity, e ∧ I + e vielbein) formulation also t of view. , it is not true in general. cted to be first class, and xpression of the potentials lism. The tetrad is a set of the ghost-free theory is not n the single massive gravity nstraints arising as primary tials include square roots of pse and shift functions, one responsible for removing the g of a certain antisymmetric ], the interaction terms ( s (due to the wedge product IJKL ry. Another part is expected s for the two metrics, the correct set of variables is d that the cut-off is raised to a , the analysis quickly becomes upposedly removing the ghost ]. In the metric formalism, the ity can in particular be trusted 27 ied out only recently in [ icle. Consequently, a number of ]) where it was pointed out that n and the original one in terms y means of canonical analysis. ]. 27 28 s of the tetrads 29 , , ǫ L − 16 ] to happen always at the linearized ]. One of the key goals of these works . e 27 26 ∧ IJ – η K − J ν 15 e e I µ ∧ e – 2 – J + ] (see also [ = e 27 A non-perturbative formulation of the action was ∧ ) between them. Linear combinations of the metric µν g I + 1 − ρν e g µρ + IJKL g ] for a reevaluation of the strong coupling issue. ( 10 V , related to the metric by µ , ǫ x L − d e I µ e ∧ K + = e I e ∧ ], including its natural description as a sector of a bi-metr J + 14 e – ∧ 11 I + e A new insight came from the work [ Modulo the above discrepancy, the tetrad (or more generally See, however, a recent paper [ , plus a local interaction argued to give a ghost-free theory. In particular, the poten 1 IJKL − ǫ µν metrics and did not seem to be natural from any geometric poin This is an importantan advance, which artificial in monstrosity, particular but showsidentified. a that The rather actual natural equivalence theory,of between once metric this variables formulatio iscontraction not exact, of but the depends tetrads. on the This vanishin was shown in [ perturbations describe a massless andpapers a massive have spin-2 studied part the proposed interaction [ two metrics, reduce to the following 4-forms The bi-gravity ghost-free interactions, rewritten in term g degree of freedom are aand second one class as pair, a with conjugaterather one secondary messy, even of constraint. in these Unfortunately the co (difficult) simpler computation vielbein of formulation the of secondary [ constraint was carr are all linear inused the in lapse constructing and them).gets shift Varying functions the with of set respect both ofrealize to metric primary the these constraints. algebra la ofto A diffeomorphisms be part of second of the class. this bi-metric In set theo particular, is the expe constraints that are s level, where the conditionConditions amounts for to its a validity were gauge recently fixing. spelled out However inmakes [ it completely transparent howghost the degree primary of constraint freedom arises. Indeed, as pointed out in [ these potentials take afour much one-forms simpler form in the tetrad forma was to establishHowever, the the ghost-freeness latter turns non-perturbatively, outand b to in be rather the cumbersome, bi-gravityV both frameworks, i owing to the complicated e given in [ with the action being the sum of two Einstein-Hilbert action much larger energy scale, so thatin the the ghost free Vainshtein massive mechanism grav region. was argued that the Vainshtein mechanism is still at play, an JHEP06(2013)068 (1.3) metric is ework of the valued in the B ] in the context of the ve gravity using what ian analysis has never hiral Plebanski actions, 27 ection replaces the spin f Plebanski formulation r the spatial metric being field. This simplifies the ase of pure gravity (with trad is a “square root” of free of the Boulware-Deser ent complication brings in in particular, we explicitly is the formalism where the itting of the local Lorentz is an independent field, like , g secondary constraint, and rs and does not produce any d in [ for a model closely related to ss effort than in the metric (or sitized anti-symmetric tensor, k γδ ivity can now be described in onsider is induced by the term could possibly expect also some ndependent proof of the absence B and a two-form s (under some assumptions to be ation of the secondary constraint ormed very efficiently, and one is nstraint. The aims of the present ly take into account the extension j ], νβ A B 34 i µα B αβγδ ˜ ǫ – 3 – ijk ǫ 1 12 ]. The first of these papers also gives an argument as = are the two-form fields for the two copies of Plebanski 30 µν − gg B − √ ]. It can be seen as a generalization of the tetrad formalism, and 33 – + B 31 replaces the tetrad one-form of the tetrad description. The B ], as well as to put the ghost-free massive gravity in the fram , where j 27 − B ∧ i + The theory we consider in this paper is given by two copies of c Thus, in this paper we study a version of the ghost-free massi However, to the best of our knowledge, the complete Hamilton B (2) chiral subalgebra of the Lorentz algebra. The SU(2) conn ij coupled together in essentially the same way as was advocate can be calledgauge a group. “chiral” description, Suchor based a without on chiral cosmological description theof constant), is chiral general and well-known spl relativity in goes [ the under c the name o gravity. We study thecompute canonical the structure algebra of ofshow the that primary theory, the constraints and system and hasghost. the seven We degrees resultin also of prove that freedom theclarified so stabilization below). that procedure Thus, it close the is results of this paper provide an i su of the theory to ain first the order Palatini framework, formulation where of the general connection relativity. the same way as ancanonically SU(2) conjugate gauge theory, to withcanonical the the analysis “triad” spatial tremendously. field Calculations fo part canable of be to work perf the out the SU(2) completevielbein) constraint gauge algebra formulation. with Secondly, much le it allows to automatical where the epsilon symbol withthat an overtilde does stands for not the need den the a metric, metric the for two-form itsseveral here definition. advantages. is Much First, its like the “cubic the phase te root”. space of This general appar relat vielbein formulation. Moreδ precisely, the interaction we c where the fundamental fields are an SU(2) connection so-called Plebanski chiral formalism. to why the stabilization procedureby of requiring the the time preserv evolution justnew fixes constraints. one of the Lagrange multiplie in the formulation St¨uckelberg in [ then constructed via the so-called Urbantke formula [ connection, and primary second-class constraints are immediatesimplifications to in see, the one structure ofpaper the elusive are secondary to co carrythe out such one first in order [ formulation analysis, been carried out in the first order formalism. Given that this JHEP06(2013)068 (2.1) o the class of rmalism. On infinite .  esponds to the second j cribed by the Plebanski ] for a reference. As we ackets of two (complete, B putation has never been is that, in the Lorentzian e we will not study these corporated in the coupled o Euclidean signature bi- have the interpretation of ion we introduce the chiral ∧ door to an 35 e main text. In particular, alism. As an illustration of i rticular the computation of d at some point supplement red Hamiltonians. rmulations mentioned above. B red Hamiltonians. This is a table reality conditions seems escribed by the following ac- g. [ ) ral description of pure general te how simple is its canonical ij existence. l structure is summarized in sub- tch. This is not attempted in the he interacting theory the problem λδ However, new issues arise after two s opposed to the only three terms of is devoted to conclusions. A number − 5 ij (Ψ 1 2 − i – 4 – F ∧ i we discuss an infinite parameter generalization which B  4.6 Z is the main part of the paper where we present the canonical Ψ] = 4 B, A, , we define the theory to be analyzed and discuss its relation t [ 3 S ), whereas the first and third ones cannot be obtained in our fo , whereas in subsection 1.2 ] 4.5 31 As will be shown below, the interaction term we consider corr The price to pay for the advantages of the chiral formulation The organization of this paper is as follows. In the next sect ). This comes from the possibility to modify the dynamics des 1.2 term in ( present work, see, however, somean comments interesting on observation is this thatto issue be the in related problem th to of theWith finding discrepancy sui this between tetrad question andmetric lying metric fo gravity. open, our results and proofs apply t signature, one has to workthe with theory complex with valued appropriate two-forms,relativity an reality these conditions. reality conditions For are thecopies well-understood. of chi general relativity are coupledof together, reality and conditions for has t to be solved essentially from scra 2.1 Plebanski formulation ofThe general chiral relativity Plebanskition formulation [ of general relativity is d appears to preserve this structure.of The appendices final section present somethe more secondary technical constraint and results, the and Poisson in bracket of pa two smea 2 Summary of the chiral Plebanski formulation Plebanski formulation of generalanalysis. relativity In and section demonstra metric formalism. Section analysis of the interacting system.section The resulting canonica action without introducing extrawill degrees argue of at freedom, thesystem end see of and e. the are paper, alsotheories such free here modifications from in can any the be details, Boulware-Deser in it ghost. is interesting Whil to note their the other hand, the useinteractions leading of to the ghost-free 2-form bi-gravity chiral( theories, formalism a opens a the power of ourwith formalism, we all use second-class ituseful constraints to exercise, compute taken as the into Poisson itthe account) br computes physical smea spatial the metric. quantityperformed that in As is any far other known as formalism. to we are aware, such a com of the ghost and moreover generalize it to the first order form JHEP06(2013)068 ge (2.3) (2.2) (2.4) (2.5) . These i ) implies A 2.5 ). The field 2.1 multiplier field . one demands the is the connection
i 3 A KL is a Lie algebra-valued IJ i eld )-valued. For Lorentzian ’s is wedge-orthogonal to ribes complexified general B σǫ not strictly follow the rule that j R ) hold. . , B are constructed as the self- ± ollows from ( condition is simply that all te reality conditions. These the fields that appear in the ction, obtaining an equation ’s be purely imaginary. The (3 i be stated as alled metricity (or simplicity) 2.5 using tetrad one-forms valued then follows as in the Palatini K J B B δ so ivita connection. ) = 0 an state the reality condition on ) Lie algebra ones, with the Lie L n ( I i . δ C k B to put an imaginary unit factor i in front , , − B
∧ (2 L J i J ∧ l repeated indices are assumed to be summed. sl e δ k B = 1 for Euclidean. Plugging the solu- field are ) should be a real metric of the required K I n in the end. ∧ ∼ B δ σ I B ) Re( 1.3 ij e ): when the δ 4 1 C , and then the reality condition ( i , i 1 3 0 a real Lorentzian signature tetrad, its complex = 2.3 – 5 – B (3 ± IJ I , = e so P j KL B = = 0 ∼ IJ i ∗ ) ∧ is its curvature, and the field ) ), one obtains the Einstein-Cartan action plus a bound- B i j (2) k , with σ⋆ also induce reality conditions on the connection B C J B A i 2.1 ( e ± in terms of su ∧ B ∧ ∧ ½ i j ( i I A A e 1 2 are two chiral projectors defined in terms of the internal Hod B ijk 3 are ǫ := , KL 2 1 2 , ± IJ + KL P i ± IJ = 1 A P and (or to itself) which gives nine real equations. In addition, i, j (2) indices are raised and lowered with the unit metric, we do = d is a multiple of the cosmological constant, and the Lagrange i , i i 2 su λ B F ) as = 0 ) back into the action ( = i for the Lorentzian signature, and C , I as σ 2.3 (4 ⋆ The reality conditions for The action given above uses complex fields and therefore desc is required to be traceless. Its variation enforces the so-c Since the When describing the Lorentzian theory, it is also convenient by passing to the Hamiltonian formulation, as shown below. so 2 3 ij i constraints Ψ conjugate is the anti-self-dual part, and both conditions i signature the conditions are quadratic in the fields, and can star 4-form obtained as the trace of the wedge product of the dual part of the two-form where reason for these conditions is clear from ( tion ( A solution to thesein constraints can be conveniently written a reality condition on theA connection. Alternatively, one c equation can be solved for signature. Such conditions canLagrangian. be given For directly metrics infields of are terms real, Euclidean of and signature thus the the connection reality and the formalism: one varies withthat respect identifies to it the in terms independent of conne the unique torsion-freerelativity. Levi-C To get aare real easiest theory, to one state has by saying to that impose the appropria metric ( ary term. The equivalence with the Einstein-Hilbert action any other Here First, one requires that the complex conjugate of any of the 3 The indices two-form. the indices one sums over should be in opposite positions. Al of the action, as to get precisely the Einstein-Cartan actio one form, algebra viewed as a vector space of complex dimension 3. The fi are obtained from the field equation for the connection that f JHEP06(2013)068 ) 4 + i 2.6 (2.8) (2.7) (2.9) (2.6) X (2.10) (2.11) (2.12) as the a ∂ a i ∼ E = , i ) upon using simply imply  X c k ij a 2.8 ∼ E b j D ∼ E . a i as n identifies ). It starts with the  ∼ E ) and ( ct to Ψ i a j bc = 0) parts, respectively, 2.1 A abc B 2.7 ǫ ∼ se space of the theory is ii a i 0 λ b volume form of the metric. owing Lagrangian B + , with conjugate momentum ) f a bi-gravity model, we recall tion ( , i a ij j k a ab . this pseudotensor. We will also use a A ∼ E F i bc λδ aints. To simplify the analysis, it ast term in the Hamiltonian being b j  B ∼ cessary condition to reproduce gen- E − of the metric. After the decomposi- c ij k a i b ∼ ∼ abc ij E ∼ E ˜ ǫ dotensors. b j +  , ∼ (Ψ 2 1 E a k i b ∼ . ∼ − ijk E = ijk i ǫ ǫ a b ijk ∼ i ) 0 ǫ ) is recovered from ( = 0 i a 2 ∼ N F abc := ǫ A ∼ + – 6 – ij i L bc 2.3 0 + b ∼ a 1 2 ∂ ij ∂ B denoting by ( i ab N = ∂ + ai ∼ F Nδ field can be omitted from now on. In terms of the i b a i ∼ i E  bc ∼ ∼ a := E E i F 0 a = a are non-dynamical. Since they enter the Lagrangian ( ai B i 0 as a N ∼ i ij 0 E B a i 0 +  Ψ B , B a i a ∼ i abc 0 E ˜ ǫ a 2 1 ,B D i 0 i 0 = A A L + i a describe its trace, antisymmetric, and tracefree ( A 0 ij ∂ b ∼ a i , is the covariant derivative defined by the connection i ∼ E b ∼ , and we omitted a total derivative term. This representatio a ), the constraints obtained by varying the action with respe k = D ∼ N, X 2.8 j a L The tilde above the symbol keeps track of the density weight of A 4 ijk and we assumed the invertibility of where Thus, these components of the The remaining fields ǫ densitized triad determining the spatial metric,the with cosmological the l term, cubicNotice in also the that the triad covariant and expression thus ( giving the remaining fields and using the notation its (densitized) inverse. Invertibility oferal the relativity, triad perfectly is analogous a to ne tion the ( invertibility linearly, they are Lagrange multipliersis generating convenient to constr decompose usual 3 + 1 decomposition of the action which leads to the foll the above Lagrangian can be rewritten as It contains onlyparametrized one by term the spatial with components time of the derivatives, connection thus the pha tilde under the symbol to characterized negative weight pseu where As a preparation for ahere more complicated the canonical canonical analysis o analysis of the chiral Plebanski formula 2.2 Canonical analysis JHEP06(2013)068 , i a A ) (2.14) (2.13) (2.16) (2.15) (2.17) tively. N,M ( , a . Indeed, as a i  N ∼ ) ion generated E G i a are the Gauss, A + , which makes clear a ) bi b i e Poisson bracket of ∼ ∼ E E a i ( the main advantage b ∼ N,M E ∼ N,N ( ∂ , ~ by a term proportional N i 0 , − a D A lting constraint precisely i b . N re to be given by the Pois- en it is straightforward to A = 0 = ) nts, as it should be in any a ach and loop quantum grav- } } ∼ , . ∂ N gly. They are first class since ) b , om . N bi N M a y iad interpretation of ∼  , H E H c k ∼ N a , − , M∂  b ∼ φ E − H a x ∂ N N b j e function multiplying the diffeomorphism − ( b a ~ ∼ N {G 3 E ˜ δ ∼ {H a M i M x N ∼ i j N∂ b ∼ 3 E δ − − D , d b a − ) , a , δ a ∼ abc N∂ k φ ǫ Z ∼ A ( ∼ b N a M a = λ b ~ j i ∂ a N,N b = ( a a N ∼ } ∂ – 7 – ∂ E + L ) a b + a N a i 0 y ijk ). To write the algebra explicitly, we define the A ∼ G H k as a triad. ( ab E N ǫ N A a b j a i F = = N ∼ b ∼ j E ) and the standard ADM decomposition of tetrad. In E := −G 2.13 ∼ , } } E i ) := ) := ) := ) φ ) a i N − G = x b ~ G ∼ ( 2.11 M E , H i ), we observe that this quantity is given by i a ab ,  × , ~ tot N a ~ i F A ~ N,N ~ N,M N, a N ∼ ( { H ( ( E bi ( 2.16 ijk a a a {D L ∼ E {D L a D N ∼ Nǫ i x , 3 ) are identified as the usual lapse and shift functions, respec , x φ x N d ~ M 2 3 3 a φ d d ensures that all constraints are preserved under the evolut ), definition ( ], the spatial metric can be read off the right-hand-side of th ~ Z N, × N ( 5 1 38 1 2 ~ L Z Z φ 2.10 G D and := := := = = φ ~ ∼ N N N } } G ]. The constraints arising by varying with respect to 2 ~ D H M φ 37 G D , , , 1 ~ φ N The resulting Lagrangian identifies the symplectic structu 36 The constraint algebra offers a direct way of confirming the tr 5 {G {D son bracket Notice that we have shifted the initial constraint coming fr of this formulation, andity also [ the basis for Ashtekar’s appro This is the same phase space of an SU(2) gauge theory, which is two Hamiltonian constraints.constraint. Namely, From it the last should line appear of ( in th smeared constraints particular, nicely explained in [ that the spatial metric is constructed from to the Gausscoincides constraint. with the Thisverify generator is the of convenient following relations spatial because diffeomorphisms. the resu Th the constraint ( Hamiltonian and diffeomorphism constraints,they correspondin form a closed algebra under ( The above algebra where we used notations by the Hamiltonian diffeomorphism invariant theory. which is given by a linear combination of first class constrai JHEP06(2013)068 ) 6 a i α ij to ∼ E ± on, 2.5 i a (3.1) A is real. stant ers Ψ a N j of this section , +  ± i B -geometry. For − ∧ ies which reduce ces B i + ∧ i B ) as compared to the ) describes two degrees + ints). This fixes ij nstead of the two-forms At the same time, the 3.1 δ lysis, or to an analogous pies of Plebanski theory, uch simpler is the above αB + onical picture. In the case λ in ( neral relativity coupled by a are the self-dual projections + 2 e interaction breaks the total etric and can be seen to be a − ike to take advantage of, and d by requiring that the reality j ± ravity. ij − iption of general relativity given ntzian case, the spatial triad B + B (Ψ ∧ i 1 2 is the connection compatible with the − = 0, the gauge group of the theory is − i a B i ) α − ij ]. δ F − 36 is purely imaginary and the shift ∧ λ i – 8 – ∼ − − N , where Γ (3) to its diagonal subgroup, and it is this fact B i a ij − − ), which imply that K + i SO ). Thus, each Plebanski term reduces to the Einstein- (Ψ 2.2 + + i 2 1 × i a F ) discussed above. The remainder of the condition ( 2.3 − ∧ − 2.5 i = Γ + ). The simplest interaction then leads to the following acti i a B Diff as in ( A  2.1 × J ± Z -sector the variation with respect to the Lagrange multipli e denote the chirality and should not be confused with the indi ± distinguish the fields from the two sectors. The coupling con ∧ (3) ± ] = + I ± ± ± e Ψ SO , is the extrinsic curvature of the spatial slice embedded in 4 ± × i a + K ,A ± ) the indices B Diff [ 2.3 is preserved by the time evolution (generated by the constra S a i In absence of the interaction term, i.e. for Our model can easily be reformulated in terms of the tetrads i Finally, we briefly discuss the reality conditions in the can As a result, the constraints generate 3 + 3 + 1 = 7 gauge symmetr In ( . Indeed, in each ∼ E 6 ± group given by the directwhich product include diffeomorphisms of and the local symmetries gauge of rotations. each Th of the co that is responsible for a larger number of propagating modes will later on get related to the mass of the second graviton. more details on this, the reader can consult [ triad, and which distinguish the two Plebanski sectors. where the indices Cartan action for the tetrad with its own cosmological term. non-interacting theory. 3 A chiral bi-gravity model A bi-gravity theory is acertain model interaction represented term. by Our two idea copiesby is of the to ge Plebanski use action the chiral ( descr of the Euclidean signature, everything is real. For the Lore becomes the requirement that the lapse 9 + 9 = 18of dimensional freedom phase of space aHamiltonian to massless analysis a as graviton. 4-dimensional comparedanalysis one It to using which the is tetrads. usual worth metric stressingwrite It ADM a how is ana similar m this “chiral” simplicity model that of we the ghost-free would bi-metric l g is taken to be real,part which of ensures the the reality reality condition of ( the spatial m of the two-forms of The reality condition for the connection can then be obtaine be of the schematic form imposes the simplicity constraints ( B JHEP06(2013)068 ] ) ), 39 ], it 1.2 3.2 (3.2) (3.3) 29 ) as our 3.1 it is parity-odd, condition of the ), and the ghost- ) which has been 1.2 1.2 he case of Euclidean . J ulation is more subtle. − ] and related references, rturbation theory, where e o deal with the bi-gravity 11 ∧ ) and makes it real as well. , omials ( I ecently pointed out in [ 7 or the interaction term ( e tetrad formulation as both ] only for the configurations tandard reality conditions of valently, for the complexified same reality conditions which al modulo a surface term [ − ibution, not of the type ( a further difference due to the term in ( te that, as pointed out in the taken to be real and one finds ted choice, we do not consider ulation turns out to be slightly ion vanishes on configurations 3.2 e 11 elves to perform the analysis for in [ rm becomes more intricate than an signature, the situation with ∧ , 7 J + e ]. This is why we take ( ∧ . 27 I + ) holds. This fact puts our formulation e = 0 3.3 + I ] L − ν e − ). In our formulation, this term simply cannot – 9 – e ∧ µ ) comes at once. It is interesting to note that this 3.2 K − I +[ e e 3.2 ∧ in each of the two sectors. This renders each metric = i in the first term in ( J + ) then becomes purely imaginary, thus the total action e σ 2.1 ∧ 3.2 I + e IJKL ǫ 2 σ ]. It is consistent with all symmetries of the theory, except 27 ). We further comment on these issues below. ). As a result, we come to an interesting observation that the 3.3 3.3 ] it was shown that this condition can always be realized in pe Let us also make a few comments about reality conditions. In t On the other hand, the relation of our model to the metric form 27 The first contribution is exactly the symmetric interaction real Euclidean bi-metric gravity.the In reality conditions contrast, in in thein Lorentzi the presence single of gravity the case. interactionhave The te been simplest idea discussed is in to impose section the signature, as in the usual Plebanski theory, all fields can be this issue any further in thethe present Euclidean paper, and signature content when ours theory, all when the all fields fields are areprevious real complex). paragraph, (or, the Nevertheless, equi term letsatisfying spoiling us ( the no reality of the act provided one multiplies the wholethe action extra by a i factor cancels ofHowever, the i. the factor As second of f termis in not ( real.Plebanski This theory implies are that not forcase. appropriate Lorentzian While and signature it should the is be s possible modified that there t is some more sophistica real and ensures also that each of the Plebanski actions is re of the ghost-free bi-gravityof theory them on the coincide same withsatisfying footing ( the as one th originally proposed in [ does not hold true in general.different from Hence, the already metric the formulation. tetradpresence form Our of model the introduces secondbe contribution discarded, in as ( the wholecontribution structure vanishes ( provided the condition ( holds only if the tetrads satisfy the symmetry property it basically amounts to a choice of gauge. However, as it was r considered in [ free interactions between two metrics originally proposed argued to generate astarting point. ghost-free massive Notice gravity also [ the presence of the second contr interaction term takes the following form In [ The reason for this is that the equivalence between the polyn unlike the rest of the Lagrangian. JHEP06(2013)068 . ts 2.2 (4.2) (4.1) (4.3a) ). The ) takes 2.8 4.1 ( . a ) plus the above + ) and ( i ), obtain explicit ∼ transforming as a E 2.7 i a 3.1  . 2.12 η 2 ci f ( / − ) , i ∼ a E , rise to two actions of the − a b − i k b − -interacting theory, leaving a ∼ A η E or this reason it is desirable j lation. However, just like it a − i . in the tetrad formalism, it is N − Motivated by this, we do the η a. The results of this analysis e model ( ∼ − E decompose as i a abc n the fields) precisely coincides abc  + a ijk ˜ ǫ ǫ ∼ as before in ( + ± onstraint responsible for the ab- i ǫ ci A
∼ a i F E on term directly contributes to the + + + + 2 bc ∼ ] N E i b c b B := ( := η i − k + a a i a a [ − i 0 ∼ ) in a simple geometric way. E and N ∼ b D B H 2 − j 3.1 ∼ abc N + ∼ ǫ ∼ E ± , i ∼ ) − bc E ) can be done in the same way as in section ijk i a + 2 ǫ – 10 – B , η i A c a 2 3.1 ( . + 0 abc + k / ǫ ∼ i ab ∼ B
E , − i 4.5 b F a -sectors. As will be shown below, these contributions − a + ∼ j − i N α ± ∼ A ∼  E E ) = = remains a connection, with the field i α + ijk + ± a i a i are again constrained to vanish by the simplicity constrain ǫ int + a + a A A ij ( + L i abc i A ± ∼ ǫ ∼ E ± b ab + F = ∼ N :=  i a a i -fields, plus the term coupling the two sectors α ∼ A E ± . In terms of the remaining components, the interaction term = ij ± int L ) for the 2.6 The canonical form of the action is then the sum of two copies o generated by Ψ Of course, only the variable only its “diagonal” partto as the introduce symmetry variables that offollowing the will field full be redefinition: theory. adapted F to this pattern. break the “off-diagonal” part of the gauge symmetry of the non matter field under the diagonal SO(3). The curvatures interaction. Being linear in lapseconstraints and already shift, present the in interacti the the following form two traceless components 4.1 3 + 1The decomposition Hamiltonian analysis and of field theThe action redefinition first ( step is toform perform ( the 3 + 1 decomposition which gives In this section weexpressions perform for a all careful constraints, canonicalsence including analysis of the of the Boulware-Deser th secondary ghost, c are and summarized compute below their in algebr subsection 4 Canonical analysis and ghost-freeness with the condition foris the not equivalence obvious with how the toalso metric include not formu that obvious how restriction to from achieve the the start reality of ( reality of the action (for the standard reality conditions o In both sectors we can introduce the fields JHEP06(2013)068 ) a i ∼ E 4.5 (4.4) (4.5) (4.6) (4.3b) . ) or (  ) does not ˆ ) H ]. However, k a 4.3 4.3 ∼ G η 40 b and shows that + . Performing a D a massive spin 2 a C H − a A i , , 2 ∼ k ∼ b N H , 2 / η 4 / ) a + / a
d as a ) D i − ˆ constraints, as explained  ( ful to contrast the above C − 0 − c ms and differences of the k a rmalism. There one keeps bj N . ansformation ( A ct that perturbations of ). The same situation arises ∼ e linearized theory. In fact, ∼ N ly the Lagrange multipliers H G cal metric can be identified ∼ hose of H b − − j µν − bove), see e.g. [ − g + 4.5 ∼ ai a ) and the phase space variables , thereby making the diagonal i H e the redefinition ( iagonal constraints by putting a se space, and does not coincide les takes the following canonical ∼ a + + a i + 0 E − and therefore is very convenient. C ∼ ∼ A N N H a ijk + µν c ǫ k 2 β N g ∼ H := := ( := ( + b = j + . Extending this redefinition to the full ) + i i ∼ c a k E ∼ ˆ G − µν G m b a µν i ∼ i H h η ∼ b j E l ψ a ± η ∼ β H + a i i + µν ∼ klm E – 11 – , q + 3 h G ǫ ,G i c k 2  − , ψ µν φ , ∼ + / g E 2 α 4 ) b j / + k / a ab ) ∼ + − ) E i i a − F ]. Thus, disentangling the relation between the physical a i η − − 0 λ + µν 0 2 ∼ )( N E 3 ∼ A 41 g N ∂ bj +   ai + + ∼ = H , nor with the ones defined in ( a 2 λ i ∼ H + a i + + ai + 0 µν ∼ + E ∼ + ∼ g A N N H α i a + A 0 := ( := ( := (   bj ∂ i ∼ a E ∼ ai φ N abc ǫ ∼ N ai ∼ E ∼ E  defined by ijk ( ǫ x ab 3 . For our model this calculation is performed in appendix 2 1 ijk g d ǫ 5 t is the covariant derivative with respect to the connection 1 2 + d a = Z D H = Let us now discuss the two Hamiltonian constraints which rea Before we give the explicit form of the constraints, it is use S (using “geometric averages” rather than algebraic ones as a change of variables withthe what original is metrics usually done as in fundamental the variables, metric and fo mixes on allows to disentangle thesymmetries diagonal manifest. and Distinguishing off-diagonal the diagonalhat sectors and on off-d the latter, theform total action after the change of variab field. However, there is no reason to expect that the linear tr theory, one may work with This is morally what we are doing here. In particular, we expe similar redefinition for the Lagrange multipliers describe the degrees of freedom of a massless graviton, and t in the metric formalism, seespatial e.g. [ metric (as appears inis the a algebra complicated of task, diffeomorphisms whateverfulfil formulation is this used. task, it Despit does disentangle the constraint algebra even without discussing theby coupling computing with the Poisson matter,in bracket the footnote of physi two smearedthe Hamiltonian physical spatial metricneither is with a non-trivial function in pha decouples the physical metric and the massive field beyond th upon linearization the(perturbations mass of eigenstates the) two turn initial out metrics, to be the su where JHEP06(2013)068 ) (4.7) (4.8) (4.9) k a (4.10) η , which b ollowing α D − k b , η . 2 can be iden- i ) a c k b i y , η λ/ ∼ D H  a ( b c i − k b + j D ∼ ∼ E x ∼ ) H H α ( b j a i bj 3 abc ∼ ˜ ∼ δ H ǫ ∼ ∼ E H a i j i α . δ ai ∼ 2 ijk H
b a ǫ in its canonical form. , ∼ ation that it removes from c δ k H − −  i ∼ λ H ) ed system. Since the presence ) α nstraint algebra, we can safely = + b j i a ) + k b − ∼ ificantly simplified by restricting mbination η s } η E − bj m b b ) j a a i + ∼ η y E 2 η λ D ∼ l a ( λ E . b ai j η  − ijk ∼  ∼ whose presence would imply that the E 2 1 H ǫ i b ( + λ , = 0 klm 2 η ) ∼ 3 c ǫ + k H a = x ijk ∼ ( ǫ i H D + ab + i a b j ( F , b η i k α ∼ ab ( H  { b ∼ ) + i c  a F H k i ∼ m b ∼ ∼ H )( α, β E – 12 – , β H + , η , ) are the Lagrange multipliers for the primary b j b 2 j
l a ) + c i k β i field. The full set of simplified constraints resulting ∼ 2 ∼ η , − H k b − H 3 ∼ ) + ), one remains with only one free parameter E ) ) and will be discussed in the next subsection. a ak i a η ak i ∼ λ i a b j H 4.3b y ) shows that the phase space of the theory is spanned j a ∼ = ∼ ∼ η + ∼ E ∼ E klm H H η E b 4.9 + j a ǫ − j λ a a i 4.4 4.11 η + D η abc + ∼ ijk x + E ǫ ∼ b ǫ j ( λ ijk − 3 ijk ∼ the term linear in 2 k β ǫ ab E ǫ ˜ ijk i δ b + a i  2 1 F η i j + ( H ∼ i + a δ αǫ ab E a ( b abc a i = 2 bj a F i D ǫ ∼ δ ( ∼ ( ∼ ∼ − H λ E b H i ijk b i a and carries the (pre-)symplectic structure encoded in the f a ǫ ijk = ∼ ∼ ai E E ǫ D 1 2 D h } ∼ ∼ h E H h 2 1 ) h  a a i i y ( ijk ∼ N + b ǫ j x x G x N x ψ x φ ∼ ∼ E E, η, = 3 3 3 3 3 , d d d d d ) ˆ A, H x ( Z Z Z Z Z i a A := := := := := { φ ~ ~ ψ G N N ˆ ˆ G G C C After we have made the choice ( H Lagrangian for the massive field contains a tadpole and is not where we denoted constraints which have the following (smeared) expression 4.2 Primary constraints The canonical form of theby action the ( fields canonical Poisson brackets can be identified with the mass of the The other variables introduced in ( These expressions, and the subsequent analysis,the can be parameters sign as the Hamiltonian constraint tified with the effective cosmologicalof constant the cosmological of constant the has combin set a very it little to effect on zero. the co The second restriction arises from the observ and These restrictions can be justified as follows. First, the co from this choice can be found in ( JHEP06(2013)068 , , we a A (4.12) (4.11) ) , (4.13a) (4.13b) a A ) N,M , ( , ) a ) , , ~ ~ 2 U G,F M G , φ ( ψ a + a ~ ~ ∂ N, N, a × ( ( i i a k U b η 1 ) ~ ~ i L L a i a η G ψ + η ˆ a ˆ a A G G D D b i η b i ) D . , ∼ N,M ) H ∼ = = = = ( ) ) H b j a   G,F ∼ ∼ } } } } pute the algebra of the ∼ N N b V ( , H b 2 b b φ ~ ˆ ~ a G ∂ a G i M ψ ∂  G ˆ V ˆ D c ∼ k G , D H ˆ − + ∼ ∼ G , -vanishing on the constraint − , ~ M∂ M∂ , ∼ G , it is convenient to shift the i b H 1 ~ a i b N ~ + b , it is not anymore first class: ˆ η j N η + ψ D . − − ) a b A ˆ j ∼ { G a H a ∂ ∼ {D η { E {D i ∼ ∼ b i ) ∂ M M c a i k − b i b b ∼ N,M H ∼ ( ∼ c E k ∼ H a H G,F ( b ∼ j ( E + ∼ ∼ U N∂ N∂ a b ∼ j + ( ( , H + ijk  U ) ∼ , a a ǫ E i i a    + ψ b b A i i i  a ∼ ) as well as , a a ) H A ∼ η ~ ∼ b ∂ N,G ψ i E A H b . i ( ) + ) a a ∼ i + ijk a × i E ~ ∼ L i ǫ E m , b N φ 2.16 , c ∼ c N,M k ∼ i  ˆ H c ˆ ˆ H j ( η  G G H G,F ∼ b ∼ a l a H ( abc ∼ H b – 13 – H + ∂ b a + ǫ j η b ∼ i V ∂ b j = = = 0 = a b ∼ i ∼ b V i G a a E ∼ E − E η } } − } } ∼ ∼ a G G i A H E klm a i b i a i + i a ψ ψ b ∼ ǫ G G a a i i abc E ∼ ˆ ˆ + N ) η A E ǫ G G ∼ ˆ ˆ ∼ ∼ N  E E ˆ H H a a , , ) + h G ˆ x ψ G , ,   ∂ ∂ φ ~ abc 3 a N φ b b k ~ i i ǫ ∼ ab − N abc N,M d − i G,F ∼ ∼ ). It is thus clear that they represent the generators of the ( ǫ ∼ {G E E F ( a {G a ~ {D U ( h h ~ x N η A Z U {D b ) := ) := j ˆ ijk weakly commute with all other constraints and form exactly a a 3 a a ∼ D ˆ Nψ D α ∼ 2.15 d H N N ~ αǫ x N a + i 3 + , x N x N ∼ ) Z − D ) E d + 2 ) 3 3 , N,M N,M , by a linear combination of the two Gauss constraints. Namely − G − G α ( ( φ d d 2 ψ ijk 2 a a a Z ~ ~ φ a ~ G N N ~ N,M G,F ǫ ∂ N,N ∂ ˆ ˆ ( ( × and ( C Z − Z C C U V h a α a 1 ~ ~ ~ V V L 4 ∼ G N φ φ G , D D = = − G G G H := := G x and 3 ~ ~ = = = = = 0 = = = N N d ~ N ˆ D D } } } } } } } } C Z 2 φ ψ F N N N M φ ˆ G ˆ G H G , H H H , H := , , , , , , ~ ~ G N 1 φ G ~ ψ G N ˆ φ N ˆ D ˆ G ˆ H It is now a straightforward although tedious exercise to com The remaining Poisson brackets are given by { H {D {G { { {G {D {H primary constraints. In contrastsome to of the the commutators case acquire ofsurface. contributions pure The which gravity first are class non part of the algebra is as follows, the same subalgebra as in ( and In complete analogy withconstraints the single copy ofdefine Plebanski theory The constraints where we used the notations introduced in ( usual gauge and diffeomorphism transformations. JHEP06(2013)068 a (4.15) (4.16) (4.14) , , weakly (4.13c) A ) a  i  i c c j j D N,G ∼ ∼ ( , H H a b b l d j j V η ∼ ∼ and E E +  d d i i i a d i ∼ G η ∼ , E H ) ∼ 0 H c k + + ∼ N,G E  ,  ( b j d } ≈ j a d j  itions ensuring that the ∼ a E , . ∼ U d d . Since ∼ i i H ˆ H a l ˆ D a c G G j c ∼  ∼ j H H , ∼ c ˆ k H ∼ A c ∼ d k H H − H G eir stability under evolution a ∼ ∂ ∼ H s ˆ finds − H a he Dirac’s stabilization proce- c i b − H − j G − b η j traint surface are proportional d i { ) , ∼ ∼ d generated by the total Hamilto- j H d ∼ H j + N . ∼ 0 H , E ∼ ∼ a E + 0 a 0 E l c k − + c j η N,G c j } ∼ , ( ∼ a d H ∼ i c H k ∼ c E a a k − H E } ≈ b j ∼ ∼ i ˆ E  E N V ∼  + D ~ H ˆ ∼ G b d j G b H i , ˆ + b b i j d + i ˆ H} ≈ ∂ , ∼ a H} ≈ l a D E ∼ ∼ ∼ c N , i ∼ H E E , ) ∼ G E A ψ )  E ∼ c ) c j ~ j c   k G N E − G ˆ ~ G H − abc ∼ i i ∼ ∼ d d ˆ {H  = E H H ~ { H ǫ ∼ N, abc N G, η η b b b b j j j ( N,G ( { {H , ˜ ǫ ∼ a a ψ 2 + ( ∼ ∼ ∼ 1 c j + E E F E ijk a φ φ G G } + + a ǫ a a a l i ∼ ˆ i ijk – 14 – } ˆ U G H − D G a a i ǫ ∼ ∼ b + + ∼ j E G E ˆ α H ˆ a ˜ α and G D ψ ∼ i d ˆ i d  E , H} H} ˆ , a G − G − a abc , , A i − ∼ A ~ NG G N η ǫ ∼ ) d ijk ) ~ ~ ∼ d ∼ ) G G NG a − ˆ H ǫ ~ D x G ~ ˆ ˆ G G G ˜ D D 3 φ G { {H h h N,G { { N, abc abc d G, ( ( ( ǫ + ∼ + ) := ) := 2 + 1 i 2 ~ V abc −G a + + abc ∼ φ φ ~ ~ Z Gǫ ǫ ∼ } G G } ǫ ˆ ∼ D i A G a G = ∼ α ∼ ∼ ∼ ˆ N a G N Gψ ˆ G ˆ α α H} D H} − N, N, , x x x x , , , ( ( N ) ~ tot 3 3 3 3 + G ˜ ˜ ˜ + i 2 i 1 + 4 ψ ψ ψ . d d d d ˆ + ) ˆ ˆ ˆ ) φ φ ) H G G D G α ~ φ N,G F ( Z Z Z Z ~ ~ ~ G,N G,G ~ G, U ≈ { ≈ { = ( ( ( α α α α i L L ~ ˙ a 2 4 L ˜ ˙ φ ˆ ˙ ˆ H ≈ { H ≈ { G ˆ ˙ ˆ D − H −D +4 H D +4 − = = = = = } } } } } ~ F G G G N ˆ ˆ ˆ ˆ D H H H H , , , , , ~ ~ G ψ ~ G G N ˆ ˆ G ˆ ˆ D D D { { { { {H where where we denoted commute with all primarydoes constraints, not the generate requirement any of conditions. th For other constraints one As is expected, allto contributions the non-vanishing mass at parameter the cons Since the primary constraints dodure not does form a not closed stop algebra, attime t the evolution preserves first the step constraints. andnian we The given have evolution by to is a study linear the combination cond of the primary constraint 4.3 Secondary constraint JHEP06(2013)068 A on (4.19) (4.21) (4.20) (4.18) (4.17) . q n ∼ H p n ∼ E f   m . f f ∼ , j j E , 1 ˆ ∼ ∼ i H} a i H H b − 1 m , g g j j } ∼ i Υ − ∼ H i H ˆ 1 ∼ ∼ i G } a E E ˆ G i − r r { i i ∼ , ˆ E G 1 fpq ∼ a ∼ E , H ǫ ∼ − ˆ a = d D l } + + ∆ i ˆ + { ers for the “off-diagonal” ∼ D ˆ ˆ H T i G a {  c   l , η r j ) is satisfied automatically. r } j ∼ a  E a ∼ ∼ } ˆ e of the Dirac’s procedure. H . As a result, it leads to the H
features the simple properties D ˆ . a  ′ D f j f y. This is done in appendix j ∼ b g G i , k ˆ , 4.16 s a number of monomials and is D }{ ∼ erves that the terms proportional a ∼ ∼ i H ˆ ∼ ˆ h, H E , a H H} H ∼ ) j ′ b E , abuse of notation denote the expressions ˆ , { a D − k ˆ i i − n-vanishing on the constraint surface. In a T 1 ∼ , ˆ H {H ∼ r G e j r j H ∼ + − a H ared constraints dropping the distributional j bcd { ∼ ∼ } ǫ E ∼ ∼ + E 1 } − E {H + a f j = f a i bcd j − } ′ ˆ h b ǫ ∼ ∼ k ˆ ∼ D } g a − E k + D E ( i ∼ , i ˆ , E ∼  i a ˆ  i D E b k G b a ′ ˆ k − , g G i , g ˆ ∼ i ∼ a ˆ k b H E H} D ∼ { a e ∼ ∼ H ∼ a , ˆ H E k  { ) E ˆ ) D  D a 1   ,T ∼ { – 15 – h H } ˆ a − i i 1 i r r D i ) hT }{ hT } ˆ η η { − ˆ ∼ + G a gcd T b H 1 , } ǫ ∼ − 7 ˆ e − ˆ  b D − D ˆ ) agf agf b e i , H e ˆ , } ) can be further manipulated using various identities s k ǫ ǫ ∼ ∼ D − b { j a ∼ d d , ∼ l l H ˆ ˆ E G ˆ h D j ∼ j b G D r ∼ ∼ + ( i ˆ { H H = det G , ∼ 4.21 E c c 1 l l ∼ }{ i  + H ) all non-vanishing Poisson brackets entering the expressi a j + ( ˆ ∆ = ( j ∼ ∼ b − G k }{ E E ˆ } G j ∼ } − i  ∼ , and not containing any derivatives nor dependence on the H , b ˆ H × × }{ b ˆ G k η G s i k a ˆ ˆ k , D , − H ∼ ˆ G 4.13c ∼ H ∼ , , h H { H , a a i j k a r i i  {H ˆ ∼ {H G ∼ ∼ ∼ E + H E E  ∼ {H G (  ∼ b N k }{ + j s e ∼ ∼ j  N + E + η ˆ b k G a k h , ∼ = = ∼ E E ijk ˆ = det a k α H} i ǫ a  , 4 ∼ ˜ {H e E ψ h G . The expression ( abr −  h + ǫ {H e A h − + Ψ = Ψ = cancel and the whole expression is proportional to Υ = Having calculated in ( In the following equation, the Poisson brackets with a slight ∼ N 7 -factor. where connection and eliminating the epsilon tensors, but the result contain for Ψ, one can proceed evaluating this constraint explicitl Thus, Ψ is the only secondary constraint arising at this stag and Although the result looksof very being complicated, linear the constraint in the field appearing under the integralother in words, we the consider partδ the of Poisson the brackets commutator of no the non-sme where the following result is obtained: Gauss and diffeomorphism constraints The first two equations can be used to fix the Lagrange multipli Plugging these expressions into the thirdto equation, one obs On the surface of this constraint, the last equation in ( following secondary constraint JHEP06(2013)068 . ) ∼ N ! and 3.3 1 ( ˆ − (4.24) (4.25) (4.23) (4.22) G } b ˆ D 0 , i ˆ G −{ seful to note that 1 − ) for the secondary } j ˆ G such that they weakly aints consists of , 4.18 s b as well as that under the , ˆ C D lation. Therefore, on con- } raint Ψ. However, due to a uting primary constraints. his, the stability condition is given by a “partial Dirac 1 constraints in the same way }{ D − ,B b ac bracket we mean here the , ) for the Lagrange multipliers } ′ ct computation. Nevertheless, , ful object since the final Dirac ass constraints. ˆ 0 j s D ˆ ′ G that it should be algebraic in G , C ≈ and ˆ , a { nd class constraints of the theory, 4.17 H raint crucially simplifies. ′ a ˆ i D ′ D ˆ ss G D − { } to note that Υ is proportional to the 1 D Ψ ′ N , − } −{ , s } ′ D a with the trivial smearing function 1 just G C ˆ ˆ ˆ D − H H H} H , A, { , i ~ N 1 ˆ G + {H ′ D – 16 – −{ } − { − D ) can be interpreted as an equation fixing one of } ˜ φ

Ψ , Ψ = = A, B 4.25 −G N { ′ = ss = {H can be replaced by the original constraints. D ′ tot ′ D , can be written as s } ˆ , H H ˙ C Ψ = ! } } A, B b b and { ˆ ˆ D ′ D , , analogous to the antisymmetric combination of the tetrads , H a i c i ˆ ˆ G ∼ D H ). It is immediate to see that the expression ( { b i a ∼ ˆ E }{ D j -function factor from the right-hand-side. Besides, it is u , ˆ i δ G abc are the two Hamiltonian constraints corrected by ˆ G , ǫ ∼ 0 ′ a ˆ ˆ H D = ( { , ′ s

C H = The stability procedure ends if ( In our case, the minimal set of non-commuting primary constr ′ ss . Their Dirac matrix and its inverse are given by ˆ D Poisson commute with allfor other Ψ primary takes constraints. the following Due form to t where where we used the fact that Ψ weakly commutes with to remove the partial Dirac bracket the Lagrange multipliers. A necessary condition for this is where we smeared the Hamiltonian constraint conjugate to the constraints constraint is equivalent to it is not yet thebracket final is Dirac obtained bracket. from Nevertheless, itas it using it is the is a missing constructed use non-commuting from Poisson bracket using all second cl The next step isits to complicated expression, study it the issome stability important difficult of conclusions to can do the be thisbracket” secondary made by of if const a the one dire two realizesDirac Hamiltonian that Ψ constraints. bracket constructed By using partialSince the generically Dir this minimal is set just a of subset non-comm of the full set of seco which must vanish for havingfigurations agreement satisfying with this the condition, the metric secondary formu const 4.4 Stability condition for the secondary constraint the total Hamiltonian, after plugging in the expressions ( combination not particularly enlightening. Finally, it is interesting where D The partial Dirac bracket is the standard Dirac’s formula JHEP06(2013)068 . ) α 4.29 ). In ) can (4.27) (4.26) (4.29) (4.28) eter = 0 and 4.29 , 4.23 , and it is a Y ∼ G ∼ H ∼ N y, one obtains and does not contain ) = ∼ N Y ′ D } of G straints and thus first 1 + ˆ a tions in phase space for . H are non-vanishing, hence der the evolution, which ∂ 0 , a rns out that this property his provides the complete ˆ rise due to our incomplete N Y though for a generic phase ≈ d the conditions ( , X , = 0, evaluate explicitly the ( he stability condition to the Y the most general form of the e second contribution. A sim- ′ D {H onal degrees of freedom at the Y ondary constraint Ψ is second ic case. ) if one notice that ( } a ∼ ∼ st some subsectors where ( N N ∼ Ψ H , ≈ + 4.25 1 ˆ ′ D ∼ N H } a { . ) as consistent additional constraints. ∂ ′ D a } = N X = 0 ˆ 4.29 Y H ˆ = Y , . As a result, the Dirac’s procedure stops . Thus, we proved that the stability condi- 1 ∼ G ∼ ′ D G ′ D } } – 17 – , + {H 1 ) in the first term and the fact that any Dirac ′ D 0 , ˆ Y H ˆ and ˆ H ≈ , H} ∼ N { 4.26 , ′ D 1 ∼ N {H + } Ψ we show that such a possibility is indeed realized and ′ D {H , , } 1 B ′ D N ) vanish simultaneously on the constraint surface, ) of the secondary constraint is a linear algebraic relation {H ˆ H} {H , 4.28 = 4.28 N Y {H , restricted to have a constant curvature set by the mass param 1 . A common solution is shown to exist in the sector with a a are functions on the phase space. Then, using Jacobi identit {H ∼ A H Y = ′ D } and ) for the secondary constraint does not contain derivatives Ψ a , X , i.e. it should not contain their spatial derivatives. It tu N 4.25 ∼ G One may consider however the possibility of having configura For generic configurations in phase space, the functions {H tion ( However, the subsectors thusmakes found impossible appear to to interpret be the not conditions stable ( un derivatives of the smearingcommutator functions. Indeed, let us write the connection secondary constraint in thelinear quadratic order in approximation, and t It is likely thatunderstanding they of do the decoupling not of havenon-linear the a level. diagonal physical and significance off-diag and a holds. Inthere fact, are in common appendix solutionsparticular, to we the study total set a of perturbative constraints expansion an around between the Lagrange multipliers ilar computation can bebe done equivalently for rewritten the as second Ψ term = in ( bracket with a second class constraint vanishes to remove th class conjugate to onedescription of of the the canonical two structure of Hamiltonian our constraints. model inwhich T gener the two factors in ( at this point, there are no tertiary constraints, and the sec follows just from the fact that Ψ is well defined, i.e. that and where at the second step we used ( where If this isclass, the further case, reducing Ψ thespace becomes configuration number weakly this of does commuting degrees not with happen, of there other freedom. still con can exi Al of the form the stability condition ( JHEP06(2013)068 9 = 36 ft func- × , with the ′ G he 5 degrees y the second ˆ H + ′ N H me important differ- eory possesses 15 con- auge symmetries of the metric part of the “off- m can be identified with and etries is broken down to the interaction term just a it is linear in the two lapse ies being two copies of the ure of this model with the aining 8 constraints are of ved 3+3 gauge symmetries, D us, even though by breaking er case, one starts with two ees of freedom. If, however, metries of the two uncoupled l subgroup by the interaction ). These constraints generate describe 7 degrees of freedom, , g to the Boulware-Deser ghost culation in appendix C shows less graviton is encoded in the ations is broken, there are still i ary constraints for off-diagonal s constraints (one primary and general relativity: SU(2) gauge G acket between two Hamiltonians in terms of our basic variables. ns. The primary constraints for st class constraint, which represents a 4.28 y of them can be chosen as second class. 7+8 = 22 variables in the initial 4 depending which of them does not commute × – 18 – H or ˆ H ). As anticipated, the identification of the physical fields B related by the condition ( , and as a result 2+2 propagating degrees of freedom. After ∼ G Diff and ) shows, the interaction term is linear in both, lapse and shi ∼ N , Ψ and either 4.2 a ˆ D ) sector supplemented by 7 first class constraints, whereas t , i ) sector, with the antisymmetric and trace parts being fixed b ˆ G ∼ E,A ∼ H, η Altogether these constraints fix 2 8 A similar mechanism is at play in our model as well, but with so Expanding around a bi-flat background, the degrees of freedo It is interesting to compare the resulting canonical struct If they both do not commute with the secondary constraint, an 8 copies of Plebanski theory isterm. also broken However, as down to ( the diagona ences. Like in the metric formulation, the group of gauge sym of freedom ofdiagonal” the ( massive one are carried by the traceless sym functions, in addition one generatesone secondary) a removing couple one of degree second ofand freedom clas leaving correspondin us with 7 degrees of freedom. the two copies ofthe gravity diagonal are group coupled, of theoriginal diffeomoprhisms. group uncoupled of theory, This and gaugethe removes thus symm interaction adds 4 term 4 of is propagating tuned the to degr g be of a special form such that with Ψ. a massless and a massive“diagonal” spin-2 ( fields. Specifically, the mass SO(3) rotations then turn outthe to spatial form diffeomorphisms a and second-class SO(3)we pair. rotations have Th we only have added remo 3 degrees of freedom. To say it differently, tions. Thus, while the off-diagonalprimary group constraints of corresponding gauge transform tooff-diagonal these spatial transformatio diffeomorphisms together with the prim uncoupled copies of gravity,group with of the diffeomorphisms group of gauge symmetr at the non-linear levelthat is the more physical spatial complicated. metric identifiedis An by quite the explicit Poisson a cal br non-trivial function lacking a simple expression one of bi-metric gravity in the metric formalism. In the latt second class: the local symmetries of thetransformations, chiral spatial Plebanski formulation and of time diffeomorphisms. The rem general ambiguity in the Dirac’s approach. This is because they can be related to each other by adding a fir dimensional phase space. As aand result, it the theory is turns free out from to the Boulware-Deser ghost. Let us summarize whatstraints. we Among have them found there so are far. 7 first Generically class the constraints: th 4.5 Summary class constraints (see appendix Lagrange multipliers JHEP06(2013)068 ], 27 ) as a but the 3.3 9 pectation 9 = 36 real × s well, 2 fields. However, × fields are imposed. 3 = 4 dimensional B is in fact first class. B × the metric formulation ˆ 2 D help of some Lagrange l of the Boulware-Deser ). The natural question e space, characterized by ns because there are too − ce oneself that this is not en by 2 3.3 The counting of degrees of 8 ach, as compared to remov- early describes two degrees ke into account that in the . In fact, the existence of nd massless graviton. Thus, the degenerate sector of the : Im del to the metric formulation anonical structure is not an − n ( ms of the n and is based on the existence r tetrad formulation of general rst time. second class. As a result, they on a more sophisticated choice ) supplemented by the standard ure was outlined already in [ ical theory with no local degrees one can allow degenerate tetrads he symmetry condition ( 18 t such a system describes just two ) also appears if one requires the ) coincides then with the real part 3.1 − 3.3 3.3 ), where this classification of constraints ) supplemented by 18 reality conditions 4.29 H, η and Ψ. However, it can be checked that 5 and ReΨ, coincide with the reality conditions ˆ – 19 – H ˆ ] for a model of non-linear massive gravity in St¨uckelberg , H a 22 ˆ D , the condition ( 3 , , Im i i ˆ G ˆ G ) once the standard reality conditions on the 3.1 ) should arise in this case as a dynamical constraint. This ex 3.3 ]. In principle, something similar might happen in our case a 8 = 16 constraints 42 × . However, the full canonical analysis of the action ( As we have already remarked above, our model is equivalent to Finally, we have also observed that there are sectors in phas Similar conclusions have been made in [ ˆ 9 D As a result, after imposing all constraints, we get 36 of these constraints, namely Re phase space corresponding toimposing the the degrees standard of reality freedom conditions, of and generating a t seco on the surface ofreal other theory constraints. not all Moreover, of one the should above ta constraints are second class and by 2 only on the configurations satisfying the symmetry conditio dimensional phase space of complex fields ( vanishing of the two (partial Dirac) brackets ( as we already discussed in section It hints that ( formalism. relativity, and as well in thewith Plebanski the formulation, where vanishing determinantnon-chiral of Plebanski the formulation metric. corresponds toof For a freedom instance, topolog [ may fail, inspecial particular, sectors leading in tounusual phase situation. fewer space It degrees with happens, of for a example, freedom drastically in the different first c orde dynamical constraint, does not provideof a way massive to gravity. relate our If mo of such reality a conditions. relation exists, it should rely massless gravitons coupled bymany a constraints gauge and fixing thefreedom massive term. in graviton modes This this disappear. happe caseof works freedom, as so follows. we The discuss diagonal only sector the cl off-diagonal part. It is giv reality of the action ( indeed turns out to beof true since the spatialreality part conditions of leads ( to a disappointing conclusion tha is then ifmultiplier one terms can added impose topossible, this the because condition action. this by It condition hand, is cannot i.e. not be with hard rewritten the to in convin ter in this paper we have given its explicit realization for the fi now remove a “half” ofing the one configurational when degree of they freedomghost are proceeds e first in the class. way analogousof On to the the the additional metric other formulatio second hand, class the pair. remova While very similar pict converted the corresponding constraints from first class to JHEP06(2013)068 ] ) ], ), 27 13 3.1 3.2 , ). The 12 ± , 8 (Ψ , ntraction of 7 ± ). This might λ ± ibes a modified (Ψ ± ) into an arbitrary dom. Such special λ 2.1 spoil it. Moreover, the hrough for both sectors ion introduces unphysical zation of our model ( ]. It can be argued to be nly 2 degrees of freedom. bi-metric gravity used here relativity is that it allows ormulations. The former is neral relativity like higher- proved that the theory only ganized as to not propagate ion of gravity, with indepen- 43 en tetrads considered in [ e formulations have more of on for general relativity, plus nt in ( ary functionals nalysis ensures that the scalar massive (bi-)gravity. The the- nteraction terms [ -Deser ghost. Therefore, even hich indicates the presence of that there is in fact an infinite to the sum of two terms ( ving the Boulware-Deser ghost ts give an independent and ex- ar. In any case, they can lead cations of the theory. We leave the functionals – 20 – ]. The specific form of the infinite summation depends on the 44 )) and the other features an alternative gauge-invariant co (Ψ). λ 1.2 (Ψ) of the Lagrange multiplier Ψ. The resulting action descr λ ) theories, which typically have additional degrees of free R ( f It is instructive to compare and contrast the formulation of Relying on these results, one can suggest a similar generali modification is achieved by turning the cosmological consta with the other, more standardthe ones most such economic as one metricredundancies, in and terms and tetrad of f both fields, the as tetrad any gauge and formulat chiral Plebanski-typ In this paper we have presentedory a we chiral considered model is for given ghost-free an by interaction two copies term. of On the shell, Plebanski acti the interaction corresponds be particularly useful concerningthe phenomenological study of appli these issues for future research. 5 Conclusions class of ghost-free bi-metric gravities, characterized by argument implying absenceof of the extra coupled degreesinteraction system, of term and freedom is the still goesthe interaction linear constraints t does responsible in not for bothwithout appear the lapses performing absence to and a of detailed shifts, the canonical w Boulware analysis, we argue dent tetrad and connection variables. (second item of ( and furthermore extend it for the first time to a first order act one of which coincides with the symmetric interaction betwe related to an infiniteextra sum degrees of of higher freedom order [ specific curvature choice terms, of or where the two cosmological constants are replaced by arbitr functional an extension toThis an should infinite be class comparedorder of with traditional actions modifications all of propagating ge o only to a reduction ofghost degrees of has freedom, been and removed therefore throughout our all a the phase4.6 space. Modified PlebanskiA theory striking property of the Plebanski formulation of general geometric interpretation of such subsectors is far from cle theory of gravity propagating two degrees of freedom only [ the tetrads. We performedpropagates a seven complete degrees canonical of analysis freedom,usually and hence plaguing effectively models remo of massiveplicit gravity. proof Thus, of our the resul absence of the ghost for a special type of i JHEP06(2013)068 ), hence onstants onditions ]. Further- 1.2 52 , st-free massive ty including the 51 works: it induces erically 8 degrees t would be an infinite art with the non-chiral etrad formalism. The bi-metric theory holds, ird terms in ( almost etrad formulations since host are non-polynomial ]. However, its canonical opriate reality conditions. onsidered in this paper is ], which is at the basis of al frame rotations without this is precisely the parity- ngly complex. The tetrad onical analysis, a fact that 50 lation, see [ his ilar to the one considered in ee interaction is interesting, ever in this case one works , preserved by time evolution. 48 s, unlike in the simple gravity s. . These in turn introduce new itions could also be related to ivalent to the metric one. itions for Lorentzian Plebanski , ebanski theories. An investiga- 49 nd term in the list. However, as ± d to not introduce extra degrees simple polynomials, but at the so opens the door to the possible re. The calculations extend im- -gravity in the metric and tetrad 47 , d if one considers Lorentz-breaking poten- 32 – 21 – ]. 55 10 class of ghost-free theories, by turning the cosmological c ). Interestingly, the term that makes the standard reality c 3.2 ] for more on Lorentz-breaking massive gravity). Here instead i 46 infinite ]. This suggests an interesting program of realizing the gho ], that can be interpreted as those of a generic massive gravi 54 53 , in contrast to the chiral case, one obtains a theory with gen 4.6 ] (see also [ 45 A way to avoid the problem of reality conditions would be to st The chiral formulation has a drawback in that the analysis of Our formalism does not allow us to incorporate the first and th into arbitrary functionals of the Lagrange multipliers Ψ An infinite number of ghost-free theories can also be obtaine 10 ± interactions for the metrics, and theof procedure freedom. can be argue The existenceand deserves of to a be more further general studied. class of ghost-fr real and Lorentzian metrics,However, the and total the action is realitycase. not conditions made One are real imaginary by these termodd condition survives extra and term spoils in ( the theory, and λ gravity as a particular subclasstion along of these modified lines (non-chiral) has Pl appeared in [ class of Lorentz-invariant potentials. the spin foam quantizationanalysis program is to of general the same relativitymore, difficulty [ modifying as the the non-chiral one Plebanski insection the theory tetrad in formu a wayof sim freedom [ scalar ghost [ fail is the same that characterizesformulations. the difference Hence, between bi findingfinding more a non-trivial reformulation reality of cond the Plebanski version totally equ Plebanski theory based on the full Lorentz group [ gravity and apply them independently to both sets of fields. T strictly speaking, only formediately metrics to with Euclidean the signatu from physical the case start of with LorentzianThe complex simplest fields, signature, possibility to how is be to supplemented borrow by the appr usual reality cond existence of an it may look more restricting, allowingwe discussed, de the facto use only of the 2-forms seco as fundamental variables al tials [ it brings onlyreduction 3 of local the gauge gaugecan rotations group be as exploited tremendously to compared simplifies perform the to a can 6 number of in explicit the calculation t in the metricformulation framework, converts which these makes complicated computationssame interactions time increasi into it introducesreally the simplifying additional the gaugea Hamiltonian symmetry chiral analysis. of model, loc which What lies we in have between c the pure metric and the t these. However, the interaction terms needed to remove the g JHEP06(2013)068 ). . , q m  (A.5) (A.1) (A.4) (A.3) (A.2) (A.6) 4.18 d l ∼ H ∼ H p m c k ∼ , , E ∼ d j d H j f l ∼ ∼ ∼ H H E bcd c j c ). The second b j l . ǫ , ∼ ∼ i E a i ∼ a E H η jkl ∼ H ǫ 4.19 bcd bcd (corresponding re- h ǫ fpq ∼ ǫ ∼ i  ǫ ∼ i b i i a j d − k b i b ∞ i ∼ ∼ E i ∼ E ∼ a , ∼ H E j E b b i c ∼ k a f stable subsectors with E i ). To evaluate them, the a i ∼ C Starting Grant 277570- ∼ dation, Germany. E ∼ H e ∼ E ∼ H a n techniques, such as loop H j = 0 ) ) −  ∼ h, = etween the diagonal and off- E e), our analysis allows us to 4.18 g more complete study of their ˆ ) j b j T gauge formulations of gravity, α ( ty. general form is given in (  ˆ e hT ∼ ∼ T tability condition for the sec- − H c E k e a a j i a i ∼ − − H ed to arrive at a conclusive result. ∼ ∼ E b ∼ − H j e H h  ∼ h  H + ( ). It can be further simplified as 2 1 + ( h + ( g j − − ∼  −  E c + k b i b e i a j ∼ a i ) E  ∼ 4.13c ∼ ∼ H b b H E j i ∼ H a i a i ∼  ∼ E hT h H ˆ ∼ ∼ H j H}  b H , − + gcd ∼ i E + ǫ + ∼ – 22 – ˆ e a a i j abc G ) in the following form b ) i b i ǫ ∼ ∼ { ∼ s E i ∼ H ∼ 1 E E e = ( ∼ a ijk i − a H i  4.18 − ǫ s i a ∼ } ∼ E 1 E i a ∂ i 2 1 ˆ  − r  G i ∼ E , h e ∼ = H a  h h ˆ e , and thus reproduces the first term in ( } D 1 1 = ∆ − i h a i − − ˆ s G k c i }{ ∼ α H , ∼ a ∆ ∆ E 4 . Using this result, one further computes ∆ a ˆ s ∆ = det ∆ j b a i D ˆ − − D ∂ − , ∼ ∆ r i H b { i a ∼ ˆ = = E α D = 1 abc = det ∼ ( E 4 { 1 ǫ ∼ 1 − ˆ h ˆ H} = − } H} abr ijk , , := } ǫ a , ∼ ǫ i ˆ a i b T 1 ˆ ˆ i a G D ∼ ˆ , E − D { , 1 {H ∆ a i i , 1 2∆ ∆ ˆ j ∼ G − E ˆ α G i a = } i }{ i ∼ ˆ a i }{ H G ˆ = det
j G = 4 , ˆ 1 , G a = e − , ˆ D T Finally, it would be interesting to see whether quantizatio Another interesting open question concerns the existence o ∆ { {H {H which allows to get the last term in ( This work was supported by contract ANR-09-BLAN-0041,DIGT, an as ER well as partially by the Alexander vonA Humboldt foun Evaluation of theIn secondary this appendix constraint we evaluate the secondary constraint whose quantization and spin foam approach, developedcan to be deal with extended to the models describing massiveAcknowledgments (bi-)gravi spectively to two decoupledexplicitly gravity characterize theories such andondary a subsectors constraint. single by They on analyzing appear unstabledynamical the properties at and s first geometric sight, interpretation but is a need term includes contributions of thefirst remaining step three is terms to in invert the matrix The first term in thisdiagonal expression Hamiltonian is constraints computed just in the ( Poisson bracket b fewer degrees of freedom. Apart from the trivial cases of and where where Its inverse can be computed as follows JHEP06(2013)068 , f l ) 3 ∼ H ctor ∼ g l H (B.1) (B.2) (B.3) (A.7) ( ∼ E O . Taking rgf a + . ǫ ∼ ∼ d i k H bc ) ∼ clude that in bc H (since the last ) ∼ c E k an be dropped, ∼ ∼ E ∼ contained in the E ∼ E H ( ∼ E b j ( bcd b ) producing the last j ∼ ǫ ∼ ) for which the sta- H ∼ i a j H r j ∼ a j A.7 E ∼ tic terms in 4.29 ∼ c E i E j a c i ∼ one subtracts the missing ) to covariant ones, which H A ∼ i a H dary constraint ensures the r j ), one can interpret the “off- s sufficient to consider sumption of non-degeneracy η i a  nt and its stability condition A.6 o impose a restriction on the b ∼ i η = 0. It should be emphasized H − tions ( ∼ j a E a nd a flat background. In partic- − 4.11 a a j i A ∼ a j H ∼ ∼ H H  ∼ ) one can show that the constraint j H ) b b i j b ∼ ∼ E , E ˆ ∼ H} b E ) i a hT i b , 2 i 4.21 ∼ i ∼ H H ∼ ˆ − ∼ G H i a H ) i a ( { η e ˆ T 1 η O = 0 e − − + ( − } b a j + − i b j  ˆ -dependent terms from ∼ 2 G ˜ H ∼ h b – 23 – i H H e , j b A j b ∼ a H can be viewed as restrictions on the antisymmetric ∼ E ˆ ∼ + ( a E i D a ). i a a i ∼ as a restriction on the trace part of  ∼ H ˆ ∆ = H }{ D ∼ b i H i a a field. Similarly, using ( ˆ i a ∼ + H 4.21 η H ˆ ) together with ( η D η b a i i , and ∼ = ∼ E + H i a i i a 4.19 ˆ are combined with similar ones from ( {H G η ∼ (2) + E η. a i η b ˆ i +  Ψ ∼ E ∼ E h
has been extensively used. As a result, dividing by the prefa a i ˆ  ) H} ∼ i 2 E , ˆ − G a ,  ∼ H ˆ ( e D (2) { ), whereas O , respectively. This observation is already sufficient to con 1 ˆ h a Ψ i . The contributions proportional to the Gauss constraints c − ∼ 1 ∼ H H η } + − i a 1 + a i a ) explicitly given in ( ∼ ˆ ∆ E η D h α , a i α and i + terms linear in 4 ∼ ˆ E ) sector we can have at most 5 propagating degrees of freedom, 4.19 G αe η − 4 }{ i = ∼ ˆ H, η ˆ G Ψ = First, let us compute the secondary constraint Ψ up to quadra , ≈ − {H Ψ from the explicit expression ( where the constraint of the first term, one concludes that in our approximation it i into account that term gives the ( parts of that this analysis is moreular, general we than will a show linearization that,of arou in the contrast physical to metric), the it later allows (under for the solutions as of the condi in a first few orders of a perturbative expansion around In this appendix we analyze in detail the secondary constrai diagonal” Hamiltonian constraint can be written as Lagrange multipliers. bility condition of the secondary constraint does not seem t term in ( B Perturbative analysis around in turn can beterms replaced linear by in the two Gauss constraints provided One can check that these terms complete the derivatives in ( whereas the terms linear in This contribution can be combined with vanishing of the trace part of the In particular, we observe that at the leading order the secon JHEP06(2013)068 . ) . 2   ∼ (B.7) (B.4) (B.6) (B.5) ) H a j c l ( (2) ∼ ∼ E H O ˆ Ψ c l ∂ ∂ ∼ + H ) d m b l 2  η ∼ d k E ∼ H k b k ∼ d E ( a k η η . (2) c j klm a ( ∼ ) O ǫ H ˆ  ∼ Ψ 2 a b H j j a ∂ D ∂ + ∼ ∼ d η D E l ), on the constraint H − , b j ( a i  = 0 can be considered ∼ ) simultaneously van- ) a H i d b j j ∼ ∼ 2 (2) O l H E d } ∼ ∼ ∼ A.5 ) + i b H E E ˆ ∼ Ψ ∼ d H E l c η B.6 k a + i ∂ ijk ( b tot j ∂ a i ∼ ǫ ∼ ∼ ints is of second class. H  e trace part is removed by ∼ E H y constraint which requires ∼ ure changes drastically. In O E l ) b E  ,H hen the Dirac’s procedure is ltipliers cancel in agreement + G ∼ E ijk  + − k d ints and Ψ become first class. ˆ c o correspond to the degrees of k ˆ Ψ k d k b H ) and ( η b j + 2 {  η ∼ η b ∼ j Gǫ H i ∼ a ( c b k E j b j ∼ H +  a E a ∼ ∼ ∼ δη H b a H E i 4.17 W D N k D a c i ∼ b j E D k b − ∼ ∼ H E ∼ + E E ( l ∼ b E + abc j a  δ η d l ǫ ∼ abc j η a + c l  ǫ b ∼ ∼ j η ∼ c H G k + ∼ b . We will compute it keeping only terms j ∼ H m b l d 1 E ∼ } E k ijk ∼ c i b o η ∼ E − E i i a η b ∼ e  a E tot a αǫ η abc η i b j a i ǫ ∼ a a i D i ∼ + ∼ E – 24 – ∼ E , appear to be non-trivial functions on the phase ∼ α ∼ E . ,H E m E c   + 3 ∼ , G − ˆ c k ∼ Ψ , − E a i ijk − G ) 4.4 m c { b l ∼ ǫ ∼ H 2 E ˆ k η ab αe ∼ b H k H l a ∼ − αe 6 and F H ∼ l d η E ijk 6 ( + in the second term, on the constraint surface (including  η   a j d ∼ k b k N a j O ∼ ∼ ∼ . To this end, using ( N G Gǫ E  ∼ klm H η ∼ E a E d j ǫ b  + b j H a b i + j ∼ ∼ ∼ N a E ∼ H d D j ∼ E ∼ E i d ∂ + H ∼ ≈ a H i − A + ~ c G j ijk ∼ + 2 E o k b ˆ ∼ D E abc η b k ∼ Gǫ ǫ ∼ a tot ijk ∼ 1 E + ǫ bcd a D − ǫ ∼ − 2 ˜ ψ b ,H i e   ˆ D  G ∼ and Ψ. ˆ b l b j E Ψ ∼ − n + N η a i ∼ n ˆ H d l H a ∼ i E a i ≈ ≈ ∼ ∼ H ∼ H E ∼ N j d a o 1 ∼ E ) and the constraint ijk − D ǫ tot e +  B.3 − ∼ ,H N = 2 ˆ Ψ = ≈ Let us now analyze the situation where both coefficients in ( Next, let us analyze the stability condition of the secondar Both coefficients, in front of H n i a ˜ W ψ G ˆ as an equation forcompleted the and corresponding Ψ Lagrange together multiplier. with one T of the Hamiltonian constra which are at most linear in In particular, all termswith with the general derivatives conclusion of of the section Lagrange mu Given the above results, a direct computation gives the vanishing of the Poisson bracket space. If at least one of them is non-vanishing, the equation where ish. One mightparticular, expect there that is in a this chance case that both the Hamiltonian constraint constra struct surface we obtain freedom of a massivethe graviton. constraints The ghost scalar carried by th symmetric traceless parts of the fields, which are expected t Using ( Ψ) this can be further simplified as JHEP06(2013)068 . ) n ˆ H ij η B.8 , (B.9) (B.8) and Ψ (B.10) (B.11) ak i ∼ ˆ E G j a η ijk = 0 so that ǫ , , . ˆ ]. Y 6 = i 27 = 0 = 0 = 0 ˆ G j a b i , ij ∼ have both coefficients ∼ E η E . It is natural to expect b onstraints which are of a a a i = 0 which automatically , = 0 ommon solution to ( D ) D ∼ E a j j i i a pace of common solutions a a c f . η η the “norm” of the field ∼ ∼ ∼ E E onsistent with the triad and H b b and the model describes one k j b b i ill removed, it would happen η round which can be described ∼ y ∼ D H abc a E ture set by the mass parameter i b appendix provides a framework ˜ ǫ ous findings [ η − D − ij a b i j i b ) ∼ ∼ η E E j a , a . a i ) which is inconsistent with our as- a ∼ i ) E . ∼ D b ∼ E E H + 2 ∼ ( H ( )) and the constraints is not empty. a b i D c ( k ijk = 0 ∼ O E D ǫ ∼ the constraints become O H − H k c j 4.29 a b = = = = = ∼ E ∼ ∼ i W E a H ˆ j a ˆ a α H G  C η + j a b j – 25 – D η j a ( ∼ E b j η i b ij ∼ b j E η η i b ∼ b a i i E η i b , ∼ ∼ E E a i η ) ∼ a i E m b ∼ E η − l a η − αe 6 αe klm , α, D α, η , ), this leads to 6 ǫ  ) k b + , and requiring that it is symmetric and traceless, solves η = 0 = 6 = 6 B.8 := := 6 a j j = 0. Using the flatness of the connection, from the Hamiltonia a ) (or more generally to ( k ab i ∼ η i ij bc ˆ ab E Y Y i a F η i a F F in ( ( B.8 ijk ) one finds that i η b a ij A i b j ǫ η ∼ ˆ ∼ ∼ E E Y being infinitesimally small. Thus, near a flat background E = + a i a 4.11 abc ∼ ij and ( E i ˜ ǫ ∼ , ab H η i a F a i ijk η H ( = 0. In the sector with vanishing δ ǫ b a i i ∼ ∼ 1 2 Y E E = = = a i First, we need to understand whether it is really possible to The above consideration suggests that one should look for a c ∼ a E ˆ Ψ = H C , rather than the flat one. Let us do this setting for simplicit that they can beto simultaneously the solved conditions which ( shows that the s ensures so that we remain with the following equations forms a second class systemmassless with and the one secondary massive constraint graviton, Ψ, consistently with previ Then we remain with 5 differential equations on 9 components o Combined with and all other constraints inα the sector with a constant curva Introducing sumption of due to an additionalsecond gauge class. symmetry, The rather perturbative expansionto than presented explicitly by in address this means these of issues. c In such situation, although the ghost degree of freedom is st The first two of theseto equations have require a the constant connection curvature, to whereas be the c fifth equation fixes To start with, we consider aby linearization around a flat backg constraint vanishing. Explicitly this requires JHEP06(2013)068 . ). ly D after (C.1) (C.2) 4.28 (B.12) cannot ∼ N ˆ Y , ) ). Our ap- ∼ ) and ( . Explicitly, H a dary constraint. ( and B.8 ∼ re the vanishing of O H Y 4.17 +  traints ( H ) to read off the physical , order in ) ,W , ∼ ability of , collecting all terms in the nts playing the role of the H ) ) k b , ffeomorphism constraint ( 2.16 η ∼ N H , we are not interested in the egenerate metrics. Therefore, ) ( O ( b ˜ j ψ ··· O ∼ ˆ + G E by the stability conditions. Thus, ) in ( simultaneously vanish. However, a + i ropriate Lagrange multipliers and their m b + ) + ∼ a E y ∼ ) ∼ G ( E D a a N − ) of the primary constraints. In this scenario, ( N,M 2 D x D ~ ( G ( a h does not intersect with the one corresponding and ) would be to insert them into the original La- 3 ˆ D lm ijk ˜ ,N δ η 1 N i ), one finds ∼ B.8 k + N y ǫ kl N η ) 3 ( η k b d j a j N ( 4.13 η ∼ V E j – 26 – G i Z a i ˆ η = ∼ H 11  E 4 m b } ) ) . + − 2 ijk c ∼ E . ∼ ∼ } ≈ f N H H a N k ) ab ( ( ∼ Gǫ y H D F H ( O O b , l ˆ . Y 1 lm ∼ c , = E + 2 . + + ) η f N l j . i x c η kl k . ( f N a ˆ i η η Y} Y} {H b ∼ j ) are those solutions which can be found from ( k , , {Y E H ) does not seem to possess an interesting dynamics and probab j ∼ E G G N η ) which are proportional to the “diagonal” lapse function a i ijk ˆ ˆ ( j H H i ǫ i B.8 ∼ E appear to be second class and affect the expression of the secon ˜ 2 ψ is ensured by fixing the Lagrange multipliers and does not requi + +  , ˆ 4.15 Y ∼ ijk ) ˆ Nη Y N N ∼ 2 G N ∼ Nǫ ( + and 2 a and Y Y ≈ − ≈ ,G ) } ≈ {H } ≈ {H N tot tot ( . However, due to ∼ G ∼ G ,H ,H ˆ The next step is to study the stability of the additional cons Y An alternative possibility to treat the conditions ( {Y { and 11 ∼ where one finds plugging in all solutions for theit Lagrange multipliers is fixed given by As a result, this scenarioto leads the to physically a interesting case constraint surface of whic massive gravity. grangian. Then theystability should appear be in studiedthe the at stability total of the Hamiltonian same step with as app the stability Our aim here is to getmetric the determined function by analogous to the diffeomorphismfull algebra. commutator, but Therefore only in the terms proportional to the di proximation allows to compute the commutators in the zeroth All terms here appear to be non-vanishing. As a result, the st be achieved unless both Lagrange multipliers the constraints Hamiltonian constraint. Suchtotal constraint Hamiltonian can ( be obtained by C Commutator of twoLet Hamiltonian us constraints consider the commutator of the two first class constrai N we are not interested inthe such sector situation defined as by it ( corresponds to d Using the constraint algebra presented in ( does not have any physical significance. JHEP06(2013)068 . ) ) b ) b (C.7) (C.4) (C.3) (C.6) (C.5) (C.8) (C.9) g 2 g (C.10) R R ,N ) + 1 + b b N f ( f . of the coupled )( ˆ G Y )( a ( ab a / g a ) g ! ∼ R . U H l g R −Y n m η ∼ + − H f + l ∼ j c = H a ∼ η a )) E f d 2 + ( j k f f , R ∼ η bmn N, ∼ N , # H ǫ g k + ( ( i d ) tric defined as a symmetric (1) R k 2 j ∼ 1 η E o  ) + ( c ∼ i ! ,G a H ∼ i b i N , ∼ ” and ”off-diagonal” variables. 1 − l g b ∼ E and expanded around a bi-flat ∼ 1 ∼ . ) = E H η ∂ 2 ) N ajk − b i η f 2 l 3 ( j ± ǫ c N αe } ∼ ∼ a E ( η ∼ a 6 E ∼ N ∼ B ab H d + j − R k ∼ f ˆ U G ) ( D + i i η − ∼ b + ∼ i , H ∼ H b O g i i k H − ∼ i 2 d c ∼ i H ˆ H ∼ G ∼ η ) ∼ E H ∼ ∼ ∼ + H N c )) b H i a )( a i b i c 2 , }{ ∼ n a d − i ∼ ∼ E i ∂ A H a E ∼ E ∼ ˆ N + ( 1 ∼ 2 E G E g  ( ) b , ∼ + ∼

i αe d N m ˆ f ( 6 a R b H  ∼ N ,G a H . One finds i 2 ( – 27 – { ) − R ∼ b a l ab a m f H + 1 ∼ ∼ G a i ∼ − E ∼ ) clearly shows that the physical metric is a very K c = H i . H N + ∼ H ( ∼ +  b a m )) H a , f b + 2( i c constructed from 2 ∼ lmn a C.7 G a ab E i f ) = ǫ ( 2 ) ∼ , it is possible to get a more explicit representation for E f N 2 k ) c a ± ( + i + ∼ i B ) + ( ∼ H

g Nf ∼ E b + V i ~ ∼ a G b ,N c b H j m ∼ ∼ H a , , g i 1 ( f H ∼ ∼ ) 1 δ H H 1 + 1 N ) = ab ) + a − i + a ( m + ( δ " 2 = } ∼ N a N E ∼ a ( − R E ab ab ( a i + ab V ˆ ) δ a b ~  ,N i ) D G ∼ ijk E 1 ∼ ( , ∼ ǫ ∼ E G E i E a ≈ N ∼ ˆ . The result ( G ∼ E )( ( + 2 + L E ( a a ( i ab ab }{ +  ∼ i ≈ V H K gg ˆ is expected to encode the spatial part of the physical metric ) G 2 , ab ab = R K ) = − R K {H 2 a i ab ∼ = E K ,N 1 a f N = ( = (1 + ( a ab in the quadratic approximation in Using results from appendix V K ab theory as The function The expression in thecombination first of line gives the the two fluctuation metrics of the me where our results imply that Let us take where Then one easily calculates that Expanding around a bi-flat background complicated function being expressed in terms of “diagonal K this further simplifies to where JHEP06(2013)068 ˆ ∼ H H t in ]. , (C.11) , ]. ommons (1972) 393 SPIRE (1972) 3368 IN SPIRE ][ , (2012) 671 B 39 IN D 6 , ]. ][ 84 , ]. SPIRE ]. but to be given (in our IN already at the quadratic . i a ][ ]. ]. Phys. Lett. Phys. Rev. η , i , ained by the constraints fluctuations of the diagonal ons either from trace of SPIRE ∂ SPIRE ]. IN arXiv:1111.2070 α IN 1 [ d reproduction in any medium, ][ SPIRE SPIRE 2 ][ hep-th/0210184 [ IN IN − Rev. Mod. Phys. , SPIRE ][ ][ = IN k j ][ ]. Effective field theory for massive gravitons ∼ gr-qc/0505134 H [ (2012) 123 (2003) 96 ajk Ghost-free massive gravity with a general Resummation of massive gravity Ghost free massive gravity in the St¨uckelberg 04 ]. SPIRE IN 305 arXiv:1107.3820 – 28 – [ ][ Strong coupling and bounds on the graviton mass in arXiv:1109.3230 , ǫ [ ]. k j SPIRE JHEP arXiv:1011.1232 arXiv:1106.3344 η [ [ , i IN [ ∂ Ghosts, strong coupling and accidental symmetries in (2005) 044003 arXiv:1007.0443 Generalization of the Fierz-Pauli Action : [ SPIRE ijk η ǫ Massive and massless Yang-Mills and gravitational fields IN [ (2012) 190 ]. α 1 Annals Phys. 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