Hamiltonian Formalism of Bimetric Gravity In Vierbein Formulation

This paper is devoted to the Hamiltonian analysis of bimetric gravity in vierbein formulation. We identify all constraints and determine their nature. We also show an existence of additional constraint so that the scalar mode can be eliminated.


Introduction
Bimetric theories of gravity are based on the idea to join the two tensorsĝ µν andf µν in a symmetric way when each tensor has its own Einstein-Hilbert action and then couple these actions through a non-derivative mass term. The presence of this term reduces the separate coordinate invariances to a single one [1,2] 2 . If we set one metric as the background metric without any dynamics we find that the bimetric theory is reduced to a single metric massive gravity theory with the mass term that at the linear limit leads to the Fierz-Pauli free theory [3]. However it was shown soon that this theory propagates ghost modes at non-linear level [4,5]. On the other hand new form of the massive term was proposed recently in [6,7,10,11] that was shown to be ghost free even at the non-linear level [12,9], see also [13,40,41].
This form of the massive gravity was further generalized in [14] where the dynamical gravity was coupled to the general reference metric. Then it was the small step to the generalization of given construction to the bimetric gravity when the fixed reference metric becomes dynamical with its own Einstein-Hilbert action [15]. It was also argued there and in [12] that this theory is ghost free. However this analysis was discussed in [34] where it was argued that the analysis performed in [12] does not show an existence of the additional constraint in case of the bimetric gravity 3 .
The non-linear massive gravity and bimetric gravity that are claimed that are ghost free are based on the specific form of the potential that contains the square root ofĝ µνf νρ . This is rather awkward structure which makes very difficult to find an extra constraint that could eliminate the Boulware-Deser ghost. However as was shown in beautiful paper [8] the square root structure suggests that the vierbein variables E A µ could be the appropriate one for the formulation of consistent bimetric theories. In more details, completely new multimetric interacting spin-2 theories were proposed in [8] using the powerful vierbein formulation of the general relativity and corresponding mass terms. It was argued there that due to the specific form of the interaction terms the action is linear in lapses and shifts which implies an existence of additional constraints that could eliminate non-physical modes. However we mean that the Hamiltonian analysis presented in given paper was not complete. In particular, the constraints corresponding to the diagonal diffeomorphism were not identified and it was not shown that they are the first class constraints.
The goal of this paper is to fill this gap and perform the Hamiltonian analysis of the bimetric gravity in vierbein formulation with the simplest form of the potential between two vierbeins E µ A and F A µ . We explicitly show that it is crucial to analyze the time developments of the constraints corresponding broken spatial rotation. It is also important to stress that when we use the parametrization of the vierbein as in [8] we should interpret p a -that will be defined below-as a dynamical variable with no time dependence in the action. As a result the conjugate momentum vanishes and is the primary constraint of the theory. Then the requirement of the preservation of given constraint leads to another secondary constraint that was not consider in the literature so far.
Very important point is to identify the constraints that are generators of diagonal diffeomorphism. To do this we follow [33] when we introduce new variables that are functions of N, N i and M, M i which are lapses and shifts inĝ µν andf µν respectively. Then we determine eight new secondary constraints where four of them should correspond to the generators of diagonal diffeomorphism on condition that the Poisson brackets between new Hamiltonian constraintR closes on the constraints surface. The similar analysis was performed previously in case of bimetric gravity formulated with metric variables in [35,36]. It turns out that in case of bimetric theory in vierbein formulation the situation is more complicated and we have to take into account the presence of the new secondary constraints. Then we are able to show that the Poisson bracket between Hamiltonian constraints vanish on the constraint surface. On the other hand one can ask the question why we use the variables introduced in [33] in case of bimetric gravity formulated with metric variables in case of the bimetric gravity formulated using vierbein formalism. The answer is that we mean that they are the only variable where it is possible to identify generators of diagonal diffeomorphism that is difficult to identify with the help of another choose of variables. Further, using this formalism we can easily see an analogy between bimetric theory formulated using either metric of vierbein variables.
With the help of this result we proceed to the analysis of the consistency of all constraints during the time development of the system. Now due to the very remarkable structure of the vierbein formulation of bi-gravity we find an existence of additional constraint which leads to the elimination of the scalar mode in the same way as in case of non-linear massive gravity [9,12]. This result confirms the results derived in [47]. More precisely, in [47] canonical analysis of bimetric gravity formulated in vierbein formalism where the spin connection is treated as an independent field was performed with elegant formulations of the secondary constraints that are responsible for the elimination of the ghosts. On the contrary our analysis is more closely related to the formulation introduced in [8] where the spin connection is not considered as an independent field however the constraints responsible for the elimination of ghost are much more complicated.
This paper is organized as follows. In the next section (2) we introduce the bimetric gravity in vierbein formalism and find its Hamiltonian, identify all constraints and determine their constraint structure. In Conclusion (3) we outline our results. Finally in Appendix we review the Hamiltonian formulation of general relativity action formulated in the vierbein formalism.

Vierbein Formulation of Bimetric Gravity
General vierbein can by written in the upper triangular form and we denote this vierbein with hatÊ (1) where N and N i are the 4 time-like components. The spatial vielbeins e a i contain 9 components that are related to the spatial part of the metric by Now by writing out the metric of this vierbein we find that means that N and N i are the usual lapse and shifts that appear in the ADM decomposition of the metric [25,26,27]. Note that the inverse metric has the form Then by definition 4Ê The upper triangular form does not fix the local Lorentz invariance since it leaves a residual spatial rotation. There are 4 components in N, N i and 9 in the spatial vielbein. The remaining 3 components of the vielbein have been fixed by using the upper triangular gauge choice. It is possible to formulate an arbitrary vierbein as the action of same boost on the upper triangular vierbein. Note that for 3−dimensional vector p a we define a standard Lorentz boost as where p a = δ ab p b and where by definition so that The boost takes the 4−dimensional vector (1, 0, 0, 0) into the unit normalized 4−vector Then we write the general vierbein as the standard boost of the upper triangular vierbein We see that 16 components of the general vierbein are now parameterized by the 4 components of N and N i together with 9 components of the spatial vielbein e a i and 3 components of p a . It is important that the Einstein-Hilbert action is invariant under local Lorentz transformation. As a result it is possible to partially fix the gauge and express the Einstein-Hilbert action using the upper triangular form. This fact greatly simplifies the Hamiltonian formalism of general relativity in vierbein formalism. The detailed analysis is performed in the Appendix A. Now we are ready to proceed to the vierbein formulation of the bimetric gravity when we consider bigravity with two metricŝ with Einstein-Hilbert actions for both of these metrics. Then without the interaction term the action is invariant under two separate local Lorentz transformations The action is also invariant under two diffeomorphisms Then we consider the action in the form [8] where µ 2 = 1 8 m 2 M 2 f g and where S n are symmetric polynomials whose explicit definitions can be found in [8]. It was shown here that they can be written in terms of traces of the matrix M as where [M] means the trace of the matrix M. In what follows we restrict ourselves to the simplest non-trivial case corresponding to β 0 = β 2 = β 3 = β 4 = 0 , β 1 = 1 which however captures the main property of given theory. Now we proceed to the Hamiltonian analysis of the bimetric theory in the vierbein formulation. We use the parametrization of the general vierbein introduced in (10). Explicitly To proceed further we use the fact that bi-gravity is invariant under diagonal local Lorentz transformation which implies that we can partially gauge fix this gauge by imposing l a = 0 [8]. Note also that since Einstein-Hilbert actions are invariant under local transformations the action depends on p a through the potential term only. Explicitly we find Using the Hamiltonian analysis performed in Appendix we find following Hamiltonian where and where π ij and ρ ij are momenta conjugate to g ij and f ij respectively. Further ∇ i and∇ i are covariant derivatives evaluated using the metric components g ij and f ij respectively. Finally note that G ijkl andG ijkl are de Witt metrics defined as that obey the relation Also note that e ≡ det e. We have also included the primary constraints L (g) ab ≈ 0 into definition of the Hamiltonian. An important point is to identify four constraints that are generators of the diagonal diffeomorphism 5 . In order to do this we proceed as in [33] and introduce following variables 5 Now due to the specific form of the interaction term we have the action that is linear in N and M and hence the first guess would be that given constraints arise as the linear combinations of R i . We checked this possibility however we found that it does not work due to the presence of the constraint k a ≈ 0 defined below. The requirement of the preservation of the constraints k a ≈ 0 led to the secondary constraints that were functions of N and M .
Then it was very difficult to identify four first class constraints that are generators of diagonal diffeomorphism. It turned out that these generators can be identified very easily using the ansatz introduced in [33] even if it was proposed for the case of bimetric gravity formulated using metric variables.
Note that their conjugate momenta are the primary constraints of the theorȳ with following non-zero Poisson brackets It is also important to stress that the absence of the time derivative of p a in the action implies following primary constraint where k a is momentum conjugate to p a with non-zero Poisson bracket We also have to identify the constraints that are generators of the diagonal spatial rotations of vielbeins e a i , f a i . These constraints are given as the linear combinations of L (g) ab , L (f ) ab and k a . Explicitly, we introduce following set of the constraints where where π i a , ρ i a are momenta conjugate to e a i , f a i respectively. Collecting all these terms together we find following form of the Hamiltonian Now we proceed to the analysis of time development of the primary constraints (25) and (27) Finally we have to the check the preservation of the constraints L diag ab ≈ 0 , L br ab ≈ 0. Firstly due to the fact that R ab according to (116) we find that they have also vanishing Poisson brackets with both L diag ab and L br ab . Then the non-zero contribution could follow from the Poisson bracket between L diag ab , L br ab andṼ. Now with the help of the following Poisson brackets and also we find that the constraint L diag ab ≈ 0 is preserved during the time evolution of the system while the requirement of the preservation of the constraint L br ab implies where we introduced new secondary constraints T ab = −T ba As we will see below the existence of these constraints will be crucial for the consistency of the theory.

Calculation of the Poisson brackets betweenR,R i
Before we proceed to the analysis of the stability of all constraints we would like to show that the Poisson brackets between the constraintsR andR i vanish on the constraint surface spanned by all constraints. To begin with we introduce the smeared form of the constraintR Then using the Poisson brackets given in (116) and following similar analysis as in case of metric formulations of bigravity we obtain [35,36] {T where and where Note also that we used the extended version of the constraintR i given in (47) and we omitted terms proportional to L (g) ab , L (f ) ab given in (116). Our goal is to show that Σ i [V] vanishes on the constraint surface. To do this we use the fact that n i can be expressed from the constraint K a where H a are functions that depend on the phase space variables whose explicit form is not important for us. To proceed further we use the fact that from the constraint T ab we derive Inserting this expression into (43) we find up to terms proportional to T ab and K a . Finally inserting this result into (42) and after some calculations we find the desired result Then collecting (40) together with (46) we find that the Poisson bracket between R is proportional to the constraintsR i , G n , S i , K a , T ab which means that it vanishes on the constraint surface. This is very important result. Note also an importance of the constraints K a , T ab for the closure of the Poisson brackets betweenR.
As the next step we calculate the Poisson brackets with the constraintsR i . However it turns out that it is more convenient to consider its following extension Let us define its smeared form Then we find following Poisson brackets which shows that T S (N i ) is the generator of the diagonal spatial diffeomorphism. Now we are ready to proceed to the calculation of the Poisson bracket between T S (N i ) defined in (48) and T T (N). In fact, using (49) we easily find Finally we should calculate the Poisson bracket between T S (N i ) and R 0 . This is really easy task using the results given in (116) so that we find up to the terms proportional to the primary constraints L (g) In the same way we can find that Using these results we are ready to proceed to the analysis of the stability of constraints.

Analysis of stability of constraints
In this section we perform the analysis of the stability of all constraints. Note that for the potentialṼ given in (33) the constraints G n , S i , K a have the form It turns out that these constraints could be simplified considerably. First of all we have following relationR so that we see that we can consider as an independent constraint following one In previous section we also found the relation so that it is possible to define new independent constraintK ĩ Then we have following set of independent secondary constraints G ′ n ,K i , S i , T ab so that the total Hamiltonian has the form Now we are ready to proceed to the analysis of the stability of all constraints. We begin with the constraints p i ≈ 0 that has the solution Ω i = 0, where Ω i is Lagrange multiplier corresponding the constraintK i . In case of p we find Further, the requirement of the preservation of L br ab takes the form where L br ab (x), G cd (y) = △ L br ab ,G cd (x)δ(x − y) and where we used the fact that Ω i = 0 together with Then it can be explicitly checked that the matrix △ L br ab ,G cd is non-singular and hence the solution of (61) is Γ cd = 0.
Using these results it is easy to analyze the requirement of the preservation of the constraints k a ≈ 0 This result however suggests to consider as an independent constraint following onẽ and following total Hamiltonian Repeating the analysis as above we find that p is trivially preserved and also Ω i = Γ ab = 0. Further, the time evolution of the constraint k a ≈ 0 is given by equation that due to the fact that f a i is non-singular implies that Γ i = 0. Finally it is also clear thatP , P i are trivially preserved. Now we proceed to the analysis of the time evolution of the constraintG n , S i ,K i together withR andR i . First of all it is easy to see that that the secondary constraintsG n , S i ,K i ,G ab are invariant under diagonal spatial diffeomorphism. Then with the help of (51) and (52) we find thatR i are preserved during the time evolution of the system.
More interesting situation occurs in case of the time evolution of the constraints G n andR which is mainly determined by following Poisson bracket and where . . . means other terms that depend on phase space variables. Note that the explicit form ofG II n , which is very complicated, is not important for us. However it is crucial and non-trivial fact that the Poisson bracket (68) does not contain terms proportional to M∂ i N or M∂ i N. Then the local form (68) has the form so that there are no derivative of the delta function on the right side of the previous equation. This fact is very important for the consistency of given theory. Now we are ready to proceed to the analysis of the consistency of the secondary constraints. In case ofG n we obtain using the fact that the Poisson bracket G n (x),G n (y) is weakly zero as follows from On the other hand the time evolution of the constraintR is equal to using (40), (46) and (51) together with the fact that Γ i = Γ ab = Ω i = 0. Now it is crucial to find non-trivial solution of (73). In case whenG II n were constant on the whole phase space we would find that the only possible solution is Γ n = 0. Then from (71) we would also find N = 0 and hence we should interpretR together with G n as the second class constraints. However this is very unsatisfactory result since it would imply the lack of the Hamiltonian constraint while the theory is manifestly invariant under diagonal diffeomorphism. FortunatelyG II n depends on the phase space variables so that it is more natural to obey (73) when we say thatG II n is an additional constraint imposed on the system. Now with this interpretation we find that (71) vanishes on the constraint surface when Γ n = 0. As the next step we will analyze the requirement of the preservation of the constraints S i , T ab ,K i which however simplifies considerably due to the fact that Γ n = Γ i = Γ ab = Ω i = 0. We start with the constraint S i using also the fact that S i (x), L br ab (y) = 0. Now due to the fact that the matrix f a i is non-singular we find that this equation can be solved for v a .
In case of the constraints T ab we find where the matrix △ T ab ,k c is defined as Now using the fact that the matrix △ T ab ,L br cd is non-singular and that v a were determined by (74) we find that the equation (75) can be explicitly solved for v ab .
Finally we proceed to the analysis of the equation of motion of the constraintK i where we defined We see that (77) can be solved for v i knowing the Lagrange multipliers v ab , v c , v n . Note that v n is still undetermined which is the reflection of the fact that p ≈ 0 is the first class constraint. Finally we should check the stability of all constraints with the constraintG II n ≈ 0 included. However it turns out that there is non-zero Poisson bracket betweenG II n ≈ 0 andG n ≈ 0 and these are the second class constraints. Then the analysis of the stability of all constraints is the same as above and we will not repeat here 6 . Let us outline our results and determine the physical degrees of freedom of given theory. We have N f.c.c. = 12 first class constraintsR,R i ,P , P i , L diag ab , p. Then we have N s.c.c. = 20 second class constraints p i , k a , L br ab ,G n ,G II n , S i ,K i , T ab . We also have N ph.s.d.f. = 58 phase space degrees of freedomN,P ,N i , P i , n, p, n i , p i , p a , k a , e a i , π i a , f a i , ρ i a . Then the number of physical degrees of freedom N p.d.f. is [39] that could be interpreted as 4 physical degrees of freedom of the massless graviton, 10 physical degrees of freedom corresponding to the massive graviton. In other words we have shown that the bi-gravity in the vierbein formulation is ghost free.

Conclusion
This paper was devoted to the Hamiltonian analysis of the bimetric theory of gravity in the form introduced in [8]. We found corresponding Hamiltonian and determined the primary constraints of the theory. Then we analyzed the requirement of the preservation of these constraints and we determined corresponding secondary constraints. Finally we determined conditions when these constraints are preserved and we found that there is an additional constraint. As a result the constraint structure of given theory suggests that this theory is free of ghosts. However it is still important to stress that even if the non-linear massive gravity is ghost free this does not mean that given theory is consistent. In fact, it was shown 6 It is necessary to stress one important point. From the form of the constraintG II n we find that G II n ,R = 0 and hence we could say thatR is second class constraint. Clearly this is rather unsatisfactory result since we would have three second class constraints while we should expect two second class constraints and one first class constraint. In order to see how to resolve this puzzle let us consider the case where the constraintsR,G n ,G II n do not depend on spatial coordinates keeping in mind that extension of this analysis to the more general case is straightforward. Note that we can use this approximation since we know that G n (x),G n (y) ≈ 0 as follows from (72).
Then the requirement of the stability of the constraintsR,G n ,G II n implies following equations ∂ tR ≈ R ,G II n λ II n = 0 that has solution λ II n = 0. Then we also find that the constraintG n is preserved. Finally the requirement of the stability of the constraintG II n implies following equation see that G n ,R ′ = G II n ,R ′ = 0 and hence we find one first class constraintR ′ and two second class constraintsG n ,G II n as expected.
that non-linear massive gravity suffers from the superluminality in its decoupling limit [16,17,18]. It was also shown that generally contain the tachyonic modes [19,20]. Further, the analysis of cosmological properties of non-linear massive gravity showed that it exhibits the ghost instabilities about its homogeneous solutions [21,22,23], see also [42,43] 7 . On the other hand it was shown very recently in [48] that non-linear bimetric theory of gravity could lead to viable cosmology under some conditions. In fact, the bimetric theory of gravity has one important advantage with respect to non-linear massive gravity when the second metric is not fixed by hand but it is dynamical as well. Clearly bimetric theory of gravity is very promising generalization of gravity that deserves to be studied further. Acknowledgement: I would like to thank S. Alexandrov for very useful discussions and for his finding of the crucial error in the first version of this paper. This work was supported by the Grant agency of the Czech republic under the grant P201/12/G028.

A Appendix: Hamiltonian Formalism of General Relativity in Vierbein Formulaton
In this Appendix we perform the Hamiltonian formalism of the general relativity in vierbein formulation. We mostly follow [28,29,30,31]. Let us consider the general relativity Lagrangian density written in the form where and where µ, ν, · · · = 0, 1, 2, 3 and where A, B, · · · = 0, 1, 2, 3. Note that by definition we have two covariant derivativesD µ and∇ µ .D µ is covariant with respect to both general coordinate transformations in spacetime as well as local Lorentz transformations on the flat index while∇ µ is covariant under general coordinate transformations. We havê We require that these covariant derivatives are compatible with the vierbein and the metricD whereĝ µν = E A µ E B ν η AB . Note that from (83) we obtain Let us now consider following 3 + 1 decomposition of tetrad where i, j, k, · · · = 1, 2, 3 and a, b, c, · · · = 1, 2, 3. The inverse vielbein obeys Using this decomposition it is rather straightforward perform the Legendre transform using this decomposition. However it is more convenient to partly break the manifest Lorentz invariance in such a way that the vierbein takes the upper triangular form (1). In this case we identify V a i with e a i where e a i defines the three dimensional metric g ij = e a i e b j δ ab . Now using (86) and also the partial gauge fixing we obtain following decomposition of Ω ABC , The general relativity Lagrangian now takes the form where e = det e a i and where we have following convention X (ab) = X ab + X ba , X (ab) X (ab) = 2X (ab) X ba .
Note that we can write where ∇ i is covariant derivative compatible with g ij so that ∇ i g jk = 0. Then neglecting the surface term we find that (89) has the form that is suitable for the Hamiltonian formulation where Note that (90) implies Ω 0(ab) Ω 0ab = 1 2 Ω 0(ab) Ω 0(ab) .
Then from (92) we find the momenta π i a conjugate to e a i π i a = δL δ∂ 0 e a Then it is easy to express Ω 0(ab) as function of π i a and e a i Ω 0(ab) = 1 e (e a i π i c δ cb − Using this result we easily find the Hamiltonian from (92) and hence corresponding Hamiltonian where R = 1 4M 2 g e (e a i π i c δ cg δ ae e e j π j g − 1 2 (e a i π i a ) 2 ) − M 2 g e (3) R , where D i is covariant derivative compatible with e a i D i e a j = 0 .
Note also that we neglected the total derivative terms in the Hamiltonian (98). It is also important to stress that (95) implies following primary constraints By definition L ab are antisymmetric so that there are three constraints L ab in three dimensions.
To proceed further we need an explicit form of D i π j a . Since π i a is the density of weight one we have 8 It is also convenient to introduce the notation π ij = 1 4 (π i a e ja + e ia π j a ) that it is similar as the notation used in [30]. Then it is easy to see that the Hamiltonian constraint R takes the familiar form where G ijkl = 1 2 (g ik g jl + g il g jk − g ij g kl ) , G ijkl = 1 2 (g ik g jl + g il g jk ) − g ij g kl , that obey the relation Note also that in the same way we can write R i = −e a i D j π j b = −2∇ i π i j (109) 8 Note that the spin connection has following prescription when it acts on object with upper and lower Lorentz indices . using the fact that − 2∇ i π i j ≈ −∇ i π i a e a j − π i a ∇ i e a j = = −∇ i π i a a a j + π i a ω a i b e b j = −e a i D j π j b (110) using also the fact that π i j = π ik g kj = 1 2 π i a e a j + L ad e ia e d j .
By definition the canonical variables are e a i and π j b with following canonical Poisson brackets e a i (x), π j b (y) = δ j i δ a b δ(x − y) so that we obtain g ij (x), π kl (y) = 1 2 (δ k i δ l j + δ l i δ k j )δ(x − y) On the other hand from (105) and from (112) we find π ij (x), π kl (y) = 1 16 (g il L kj + g jl L ki + g jk L li + g ik L lj )δ(x − y) = = µ ijkl δ(x − y) .
This result implies that there are additional terms when we calculate the Poisson brackets between the constraints as was nice shown in [30]. More precisely, let us introduce the smeared form of the constraints R, R i and L ab where We see that there are additional terms on the right side of the Poisson brackets between constraints that are proportional to the primary constraints L ab . These terms also vanish on the constraints surface. For that reason we will not write the explicit form of these terms in the calculations performed in the main body of the paper.