Is Bimetric Gravity Really Ghost Free?

0.18 MB·English

by  J. Kluson

PreprinttypesetinJHEPstyle-HYPERVERSION
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Is Bimetric Gravity Really Ghost Free?
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0
2

n
a
J

1
2
J. Klusonˇ

]
h Department of Theoretical Physics and Astrophysics
t Faculty of Science, Masaryk University
-
p
Kotl´aˇrska´ 2, 611 37, Brno
e
h Czech Republic
[ E-mail: [email protected]

2
v
6
Abstract: We perform the Hamiltonian analysis of the bimetric theory of gravity intro-
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2 duced in [arXiv:1109.3515 [hep-th]]. We carefully analyze the requirement of the preserva-
3
tionofallconstraintsandwefindthatthereisnoadditionalconstraintthatcouldeliminate
.
1
the ghost mode.
0
3
1
Keywords: Bimetric Gravity , Hamiltonian Formalism.
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i
X
r
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Contents
1. Introduction and Summary 1
2. Bimetric Gravity 3
3. Preservation of Constraints in Case of Minimal Bimetric Gravity 7
A. Hamiltonian Analysis of F(R) Bimetric Gravity 12
1. Introduction and Summary
Recently the new very interesting formulation of the non-linear massive gravity [1, 2] was
introducedwithsignificantimprovementreachedin[3,4]. Thistheorywasfurtherextended
in [5] where the theory was formulated with general reference metric. The most crucial
fact that is related to given theory is the proof of the absence of the ghosts that are
generallyexpectedinanytheorythatbreaksthediffeomorphisminvariance. Asiswellknow
the physical content is determined in the Hamiltonian formulation when all constraints
are identified together with their nature. This analysis was performed in several papers
[6, 7, 8, 9, 10, 11, 12] with the most important results derived in [13, 14] with the outcome
that this non-linear massive theory possesses one additional constraint and the resulting
constraint structure is sufficient for the elimination of the ghost degree of freedom.
Very interesting extension of given theory was suggested in [15] when the kinetic term
for the general reference metric was introduced and hence gˆ and fˆ come in the sym-
µν µν
metric way in the action. Then it was argued in [13] that the resulting theory is the ghost
free formulation of the bimetric theory of gravity 1.
The goal of this paper is to perform the Hamiltonian analysis of the bimetric theory
of gravity in the form introduced in [15]. For simplicity we call this theory as the new
bimetric theory of gravity (NBTG). We would like to explicitly determine the structure
of the constraints and eventually to prove the absence of the ghosts. Remarkably we find
very subtle issue related to NBTG which forces us to doubt whether the ghost could be
eliminated in given theory or not. More explicitly, the non-linear massive gravity with
general reference metric has the potential that depends on the matrix Hµ gˆµρfˆ where
ν ρσ
≡
fˆ is fixed background metric. There is now no doubt that such theory is ghost free.
ρσ
On the other hand the situation changes in case of the bimetric theory of gravity when we
promotefˆ asanadditionaldynamicalfieldwiththekinetictermgivenbyEinstein-Hilbert
µν
action. Then, since the interaction term between two metrics has the square structure
1Forfurther analysis of given theory,see [15, 16, 17, 18,19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29].
– 1 –
form we follow [13, 14, 15] and perform the redefinition of one shift function that makes
the theory linear in N and M which are the lapse functions in gˆ and fˆ respectively.
µν µν
It is important that the action has the same structure as the non-linear massive gravity
action with general reference metric fˆ with additional kinetic term for fˆ . However the
µν µν
fact that fˆ is dynamical has crucial impact on the Hamiltonian structure of given theory.
µν
Explicitly, theHamiltonian isgiven asthelinear combination of theconstraints as opposite
to the case of the non-linear massive gravity where the Hamiltonian does not vanish on
the constraint surface. Now the crucial point is that the components of the metric fˆ
µν
that appear as the fixed parameters in the non-linear massive gravity case should be now
considered as Lagrange multiplicators whose values are determined by the requirements of
thepreservationofallconstraintsduringthetimeevolutionofthesystem. Howeverthenwe
find that the requirement of the preservation of the constraint whose explicit definition
0
C
will be given below leads to the differential equation for the Lagrangian multiplicator M
that is related to the . In other words the value of the Lagrange multiplicator M is
D
determined by the requirement of the preservation of during the time evolution of the
0
C
system. In the same way we fix the value of the Lagrange multiplicator N. Say differently,
and could be interpreted as the second class constraints.
0
C D
At this place we should compare our result with the known proof of the absence of
the ghosts in the bimetric theory. It was shown in the very nice paper [13] that the
requirement of the preservation of the constraint implies an additional constraint
0 (2)
C C
(in their notation) given in the e.q. (3.32) in this paper. We see that this constraint
contains the covariant derivative of M which, as we argued above, is fixed in case of the
non-linearmassivegravitytheorysothatitisreallynaturaltointerpret asanadditional
(2)
C
constraint. Then this constraint together with are responsible for the elimination of the
0
C
ghost mode which is crucial for the consistency of non-linear massive gravity. However in
case of the bimetric gravity M should be considered as the Lagrange multiplicator whose
value is fixed by the consistency of given theory.
The fact that and should be interpreted as the second class constraints has im-
0
C D
portant consequence for the dynamics of the theory. More precisely, since the Hamiltonian
is given as the linear combination of the constraints implies that the resulting Hamiltonian
vanishesstrongly. Thisis ratherpuzzlingresultandwebelievethatthisisaconsequenceof
the redefinition of the shift function [13, 14, 15] which is certainly useful for the non-linear
massive gravity where the diffeomorphism invariance is explicitly broken but we are not
sure whether it is suitable for the bimetric theory of gravity where all fields are dynamical
and the theory is manifestly diffeomorphism invariant under diagonal diffeomorphism. In
fact, it is non-trivial task to identify such generator as was shown recently in [19] in case
of particular model of bimetric theory of gravity [35]. We are currently analyzing NBTG
following [19] and we believe that it is possible to identify four first class constraints that
are generators of the diagonal diffeomorphism. On the other hand the analysis performed
so far suggests that it is very difficult or even impossible to find an additional constraint
that could eliminate the additional mode 2.
2This analysis will appear in forthcoming publication.
– 2 –
We should however stress that we have to be very careful with definitive conclusions.
We wanted to show that the extension of the non-linear massive gravity to the bimetric
theory of gravity as was performed in [15] could be more subtle that we initially thought
and that it is not really clear whether given theory is ghost free or not. It is still possible
that there exists the way how to find four first class constraints that are generators of
diagonal diffeomorphism together with additional constraints that eliminate ghost mode.
Clearly more work is needed in order to resolve this issue.
This paper is organized as follows. In the next section (2) we review the Hamiltonian
formulation of the bimetric theory of gravity [15] and identify primary and the secondary
constraints. Then in section (3) we calculate the algebra of constraints for the case of
the minimal version of the bimetric gravity. Finally in Appendix A we briefly discuss the
Hamiltonian formulation of the bimetric F(R) theory of gravity.
2. Bimetric Gravity
In this section we review the main properties of the bimetric theory of gravity in the
formulation presented in [15]. The starting point is following action
S = M2 d4x gˆ(4)R(g)+M2 d4x fˆ(4)R(f)+
g − f −
Z Z q
p 4
+ 2m2M2 d4x gˆ β e ( gˆfˆ) ,
eff − n n −
Z n=0 q
p X
(2.1)
where
1 1
M2 = ( + )−1 , (2.2)
eff M2 M2
g f
andwheregˆ ,fˆ arefour-dimensionalmetriccomponentswith(4)R(g),(4)R(f) correspond-
µν µν
ing scalar curvatures. Further, e (A) are elementary symmetric polynomials of the eigen-
k
values of A. For generic 4 4 matrix they are given by
×
e (A) = 1 ,
0
e (A) = [A] ,
1
1
e (A) = ([A]2 [A2]) ,
2
2 −
1
e (A) = [A]3 3[A][A2]+2[A3] ,
3
6 −
e (A) = 1(cid:0) [A]4 6[A]2[A2]+3[A(cid:1)2]2+8[A][A3] 6[A4] ,
4
24 − −
ek(A) = 0 ,f(cid:0)or k > 4 , (cid:1) (2.3)
where Aµ is 4 4 matrix and where
ν
×
[A] = TrA = Aµ . (2.4)
µ
– 3 –
Of the four β two combinations are related to the mass and the cosmological constant
n
whiletheremainingtwocombinationsarefreeparameters. Ifweconsiderthecasewhenthe
cosmological constant is zero and the parameter m is mass, the four β are parameterized
n
in terms of the α and α of [1, 2]
3 4
1
β = ( 1)n (4 n)(3 n) (4 n)α +α . (2.5)
n 3 4
− 2 − − − −
(cid:18) (cid:19)
Theminimalaction correspondstoβ = β = 0that implies α = α = 1andconsequently
2 3 3 4
β = 3 ,β = 1.
0 1
−
Our goal is to find the Hamiltonian formulation of given theory and determine cor-
responding primary and the secondary constraints. As the first step we introduce 3+1
decomposition of both gˆ and fˆ [30, 31]
µν µν
gˆ = N2+N gijN , gˆ = N , gˆ = g ,
00 i j 0i i ij ij
−
1 Ni NiNj
gˆ00 = , gˆ0i = , gˆij = gij
−N2 N2 − N2
(2.6)
and
fˆ = M2+L fijL , fˆ = L , fˆ = f ,
00 i j 0i i ij ij
−
1 Li LiLj
fˆ00 = , fˆ0i = , fˆij = fij , Li = L fji ,
−M2 −M2 − M2 j
(2.7)
and where we defined gij and fij as the inverse to g and f respectively
ij ij
g gkj = δ j , f fkj = δ j . (2.8)
ik i ik i
Following [3, 4, 5, 15] we perform following redefinition of the shift function
Ni = Mn˜i+Li+ND˜i n˜j (2.9)
j
so that the resulting action is linear in M and N. Note that the matrix D˜i obeys the
j
equation [3, 4, 5, 15]
√x˜D˜i = (gik D˜i n˜mD˜k n˜n)f (2.10)
j − m n kj
q
and also following important property
f D˜k = f D˜k .
ik j jk i
(2.11)
Then after some calculations we derive the bimetric gravity action in the form [3, 4, 5, 15]
S = M2 dtd3xM f[K˜ ˜ijklK˜ +R(f)]+M2 dtd3xN√g[K ijklK +R(g)]+
f ijG kl g ijG kl
Z Z
p
+ 2m2M2 dtd3x√g(M +N ) ,
eff U V
Z
(2.12)
– 4 –
where
1
K = (∂ g N (n˜,g) N (n˜,g)) ,
ij t ij i j j i
2N −∇ −∇
1
K˜ = (∂ f ˜ L ˜ L ) ,
ij t ij i j j i
2M −∇ −∇
(2.13)
where
N = Mg n˜j +g Lj +Ng D˜k n˜j , L = f Lj , (2.14)
i ij ij ik j i ij
and where ,R(g) and ˜ ,R(f) are the covariant derivatives and scalar curvatures calcu-
i i
∇ ∇
lated using g and f respectively. Further, ijkl and ˜ijkl are de Witt metrics defined
ij ij
G G
as
1 1
ijkl = (gikgjl+gilgjk) gijgkl , ˜ijkl = (fikfjl+filfjk) fijfkl (2.15)
G 2 − G 2 −
with inverse
1 1 1 1
= (g g +g g ) g g , ˜ = (f f +f f ) f f (2.16)
ijkl ik jl il jk ij kl ijkl ik jl il jk ij kl
G 2 − 2 G 2 − 2
that obey the relation
1 1
klmn = (δmδn +δnδm) , ˜ ˜klmn = (δmδn +δnδm) . (2.17)
GijklG 2 i j i j GijklG 2 i j i j
Finally, and introduced in (2.12) have the form
V U
= β +β √x˜D˜i +β 1√x˜2(D˜i D˜j D˜i D˜j )+
V 0 1 i 22 i j − j i
+ 1β √x˜3[D˜i D˜j D˜k 3D˜i D˜j D˜k +2D˜i D˜j D˜k] ,
6 3 i j k − i k j j k i
= β √x˜+β [√x˜2D˜i +n˜if D˜j n˜k]+
U 1 2 i ij k
+ β [√x˜(D˜l n˜if D˜j n˜k D˜i n˜kf D˜j n˜l)+ 1√x˜3(D˜i D˜j D˜i D˜j )]+β √f ,
3 l ij k − k ij l 2 i j − j i 4√g
(2.18)
where
x˜ = 1 n˜if n˜j . (2.19)
ij
−
The action (2.12) is suitable for the Hamiltonian formalism. First we find the momenta
conjugate to N,n˜i and g
ij
π 0 , π 0 , πij = M2√g ijklK (2.20)
N ≈ i ≈ g G kl
together with the momenta conjugate to M,Li and f
ij
ρ 0 , ρ 0 , ρij = M2 f ˜ijklK˜ . (2.21)
M ≈ i ≈ f G kl
p
– 5 –
Then after some calculations we find following Hamiltonian
H = d3x(πij∂ g +ρij∂ f ) =
t ij t ij
−L
Z
= d3x(N +M +Li ) ,
0 i
C D R
Z
(2.22)
where
1
= πij πkl M2√gR(g)+ (g)D˜kn˜l 2m2M2 √g ,
C0 M2√g Gijkl − g Rk l − eff V
g
1
= ρij˜ ρkl M2 fR(f)+n˜i (g) 2m2M2 √g ,
D M2√f Gijkl − f Ri − eff U
f
p
(g) (f)
= + ,
Ri Ri Ri
(2.23)
where we also denoted
(g) = 2g πlk , (f) = 2f ˜ ρlk . (2.24)
Ri − ik∇l Ri − ik∇l
From previous analysis we see that we have eight primary constraints
π 0 , π 0 , ρ 0 , ρ 0 . (2.25)
N i M i
≈ ≈ ≈ ≈
Thenthenextstepistoanalyzetherequirementthattheseconstraintsarepreservedduring
the time evolution of the system
∂ π = π ,H = 0 ,
t N N 0
{ } −C ≈
∂ ρ = ρ ,H = 0 ,
t M M
{ } −D ≈
δ(D˜k n˜j)
∂ π = π ,H = Mδk +N j 0 ,
t i { i } Ck i δn˜i ≈
!
∂ ρ = ρ ,H = 0 ,
t i i i
{ } −R ≈
(2.26)
where
n˜pf
= (g)+2m2M2 √g pm[β δm+β [δmD˜l D˜m]+
Ci Ri eff √x˜ 1 i 2 i l− i
+ β √x˜2(1δm(D˜n D˜p D˜mD˜n )+D˜mD˜l D˜mD˜n )] ,
3 2 i n p− n m l i − i n
(2.27)
and where we used the canonical Poisson brackets
N(x),π (y) = δ(x y) , n˜i(x),π (y) = δiδ(x y) ,
{ N } − j j −
1
g (x),πkl(y) = (δkδl +δlδk(cid:8))δ(x y) , (cid:9)
ij 2 i j i j −
nM(x),ρ (y)o = δ(x y) , Li(x),ρ (y) = δiδ(x y) ,
{ M } − j j −
1
f (x),ρkl(y) = (δkδl +δlδk(cid:8))δ(x y) (cid:9)
ij 2 i j i j −
n o (2.28)
– 6 –
and also following important relations [13]
δ√x˜D˜i 1 δ(D˜mn˜p)
i = n˜nf p ,
δn˜i −√x˜ nm δn˜i
∂ δ(D˜k n˜n)
Tr(√x˜D˜)2 = 2n˜pf D˜m n ,
∂n˜i − pm k δn˜i
δ δ(D˜p n˜n)
Tr(√x˜D˜)3 = 3√x˜n˜kf D˜mD˜n n .
δn˜i − km n p δn˜i
(2.29)
In summary we have following 16 constraints
primary : π 0 ,π 0 ,ρ 0 ,ρ 0 ,
N i M i
≈ ≈ ≈ ≈
secondary : 0 , 0 , 0 , 0 .
0 i i
C ≈ D ≈ C ≈ R ≈
(2.30)
Now we have to check the stability of all constraints when the total Hamiltonian takes the
form
H = d3x(N +M +Li +uNπ +uiπ +
T 0 i N i
C D R
Z
+ vMρ +viρ +Σi ) ,
M i i
C
(2.31)
where N,M,Li,uN,ui,vM,vi,Σi are Lagrange multiplicators related to the constraints
(2.30). For simplicity we restrict ourselves to the case of the minimal bi-metric theory.
3. Preservation of Constraints in Case of Minimal Bimetric Gravity
The minimal bimetric theory is defined by the following choice of parameters
β = 3 ,β = 1 ,β = 0 ,β = 0 ,β = 1 . (3.1)
0 1 2 3 4
−
Now we proceed to the analysis of the preservation of all constraints given in (2.30). It
is easy to see that the constraint π 0 is trivial preserved. On the other hand the
N
≈
requirement of the preservation of the constraint π 0 takes the form
i
≈
∂(D˜k n˜j)
∂ π = π ,H = Mδk + j + d3xΣj π , (x) = 0 ,
t i { i T} − i ∂n˜i Ck { i Cj }
!
Z
(3.2)
where
1 n˜kf n˜lf
ki lj
{πi(x),Cj(y)} = "√x˜fij + √x˜3 #δ(x−y) ≡ △πi,Cjδ(x−y) .
(3.3)
– 7 –
Since
f 1 1
det△πi,Cj = det √ixk˜ det δkj + x˜n˜kn˜mfmj = x˜5/2 detfij 6= 0
(cid:18) (cid:19) (cid:18) (cid:19)
(3.4)
we findthat △πi,Cj is non-singular matrix on the whole phase space. However this fact also
implies that the equation (3.2) has trivial solution
Σi = 0 . (3.5)
In the samy we proceed with the analysis of the time evolution of the constraint
i
C
∂ (x) = (x),H = d3y(N(y) (x), (y) +
t i i T i 0
C {C } {C C }
Z
+M(y) (x), (y) +vj(y) (x),π (y) = 0 .
i i j
{C D } {C }
(cid:1) (3.6)
According to (3.4) we see that (3.6) can be solved for vi as functions of the canonical
variables and N,M. Say differently, π and are the second class constraints.
i i
C
As the next step we consider the constraint . It turns out that is convenient to
i
R
extend it by the expression ∂ n˜jπ +∂ (n˜jπ ) and consider its smeared form
i j j i
T (Ni)= d3xNi( (g) + (f)+p ∂ φ+∂ n˜jπ +∂ (n˜jπ )) d3xNi ˜ . (3.7)
S Ri Ri φ i i j j i ≡ Ri
Z Z
Then using the canonical Poisson brackets we find
T (Ni),g (x) = ∂ g (x)Nk(x) ∂ Nk(x)g (x) g (x)∂ Nk(x) ,
S ij k ij i kj ik j
− − −
(cid:8)TS(Ni),πij(x)(cid:9) = ∂k(Nk(x)πij(x))+∂kNi(x)πkj(x)+πik(x)∂kNj(x) ,
−
(cid:8)TS(Ni),fij(x)(cid:9) = ∂kfij(x)Nk(x) ∂iNk(x)fkj(x) fik(x)∂jNk(x) ,
− − −
(cid:8)TS(Ni),ρij(x)(cid:9) = ∂k(Nk(x)ρij(x))+∂kNi(x)ρkj(x)+ρik(x)∂kNj(x) ,
−
(cid:8) TS(Ni),n˜i(x)(cid:9) = Nk(x)∂kn˜i(x)+∂jNi(x)n˜j(x) ,
−
(cid:8)TS(Ni),πi(x)(cid:9) = ∂k(Nkπi)(x) ∂iNk(x)πk(x) .
− −
(cid:8) (cid:9) (3.8)
Then we easily find the familiar result
T (Ni),T (Mj) = T ((Nj∂ Mi Mj∂ Ni)) . (3.9)
S S S j j
−
(cid:8) (cid:9)
To proceed further we need to know the Poisson bracket between T (Ni) and D˜i which
S j
can be determined when we know the explicit form of D˜i [3, 4, 5, 15]
j
D˜i = gikf Qm(Q−1)n ,
j km n j
q (3.10)
– 8 –
where
1
Qi = x˜δi +n˜in˜kf ,(Q−1)j = (δj n˜jn˜mf ) . (3.11)
j j kj k x˜ k − mk
Using the explicit form of D˜i given in (3.10) we see that the Poisson bracket between
j
T (Ni) and D˜i is determined by the Poisson brackets between T (Ni) and g ,f and
S j S ij ij
Qi . The Poisson brackets between T (Ni) and g and f were given in (3.8) and the
p S ij ij
Poisson bracket between T (Ni) and Qi can be easily determined using (3.8) and (3.11)
S j
T (Ni),Qi = ∂ Qi Nk +∂ Nin˜kn˜mf n˜in˜mf ∂ Nk =
S j − k j k mj − mk j
(cid:8) (cid:9) = −∂kQijNk +∂kNiQkj −Qik∂jNk . (3.12)
Then with the help of this result we find
T (Ni),D˜i = ∂ D˜i Nk +∂ NiD˜k D˜i ∂ Nk (3.13)
S j − k j k j − k j
n o
and finally collecting all these results we obtain
T (Ni), = ∂ Ni ∂ Ni ,
S 0 i 0 i 0
C − C − C
(cid:8)TS(Ni), (cid:9) = ∂i Ni ∂iNi ,
D − D − D
(cid:8)TS(Ni), i(cid:9) = ∂jNj i Nj∂j i ∂iNj j .
C − C − C − C
(cid:8) (cid:9) (3.14)
Then it is easy to see that T (Ni) is preserved during the time evolution of the system
S
and that it corresponds to the generator of the spatial diffeomorphism. In other words ˜
i
R
are first class constraints.
Now we come to the calculation of the Poisson brackets between the constraints
0
C
and . It turns out that it is useful to introduce the smeared form of these constraints
D
C(N) = d3xN(x) (x) , D(M) = d3xM(x) (x) . (3.15)
0
C D
Z Z
We begin with the Poisson bracket between (x) and (y). Since does not depend on
0 0 0
C C C
ρij we immediately find that the Poisson bracket between (x) and (y) has the same
0 0
C C
form as in [13] which means that it vanishes on the constraint surface
(x), (y) 0 . (3.16)
0 0
{C C } ≈
In case of we find
D
D(M),D(N) =
{ }
= d3x(∂ MN ∂ NM)[fij( (f)+ (g))+(n˜in˜j fij) (g) +n˜i2m2M2 √g√x˜]=
i − i Rj Rj − Rj eff
Z
= d3x(∂ MN ∂ NM)[fij +(n˜in˜j fij) ] .
i i j j
− R − C
Z
(3.17)
– 9 –