Conformal Traceless Decomposition of Lagrange Multiplier Modified Horava-Lifshitz Gravity

0.14 MB·English

by  J. Kluson

PreprinttypesetinJHEPstyle-HYPERVERSION
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Conformal Traceless Decomposition of Lagrange
1
0
2 Multiplier Modified Hoˇrava-Lifshitz Gravity

b
e
F

6

]
h
t J. Klusonˇ
-
p
e Department of Theoretical Physics and Astrophysics
h Faculty of Science, Masaryk University
[
Kotl´aˇrska´ 2, 611 37, Brno

2 Czech Republic
v
E-mail: [email protected]
6
2
7
7
Abstract: We introduce conformal traceless decomposition in Lagrange Multiplier mod-
0
. ified RFDiff invariant Hoˇrava-Lifshitz gravity. We perform Hamiltonian analysis of given
1
0 action and determine the action for the physical degrees of freedom.
5
1
: Keywords: Hoˇrava-Lifshitz gravity.
v
i
X
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Contents
1. Introduction and Summary 1
2. Brief Review of Lagrange Multiplier Modified HL Gravity 3
3. Conformal Traceless Decomposition 4
1. Introduction and Summary
In 2009 Petr Hoˇrava formulated new proposal of quantum theory of gravity (now known as
Hoˇrava-Lifshitz gravity (HL gravity)) that is power counting renormalizable [1, 2, 3] that
1
is also expected that it reduces do General Relativity in the infrared (IR) limit . The
HL gravity is based on an idea that the Lorentz symmetry is restored in IR limit of given
theory while it is absent in its high energy regime. For that reason Hoˇrava considered
systems whose scaling at short distances exhibits a strong anisotropy between space and
time,
x′ = lx , t′ = lzt . (1.1)
In order to have power counting renormalizable theory we have to demand that z 3 in
≥
(3+1) dimensional space-time. It turns out however that the symmetry group of given
theory is reduced from the full diffeomorphism invariance of General Relativity to the
foliation preserving diffeomorphism
x′i = xi+ζi(t,x) , t′ = t+f(t) . (1.2)
Due to the fact that the diffeomorphism is restricted (1.2) one more degree of freedom
appears that is a spin 0 graviton. The existence of this mode could have very significant
−
consequences either for the consistency of given theory or for the phenomenological appli-
cations of HL gravity. For that reason it would be desirable to formulate HL gravity where
the number of the physical degrees of freedom is the same as in case of General Relativity.
Such a proposal was formulated by Hoˇrava and Malby-Thompson in [15] in the context of
2
the projectable HL gravity . Their construction is based on an extension of the foliation
preserving diffeomorphism in such a way that thetheory is invariant underadditional local
U(1) symmetry. The resulting theory is known as non-relativistic covariant theory of grav-
ity. It was shown in [15,16] that thepresence of this new symmetry implies that the spin-0
graviton becomes non-propagating and the spectrum of the linear fluctuations around the
background solution coincides with the fluctuation spectrum of General Relativity.
1Forreview and extensivelist of references, see [4, 9, 10, 11].
2Seealso [23] and [24]
– 1 –
It is also well known that General Relativity contains large number of symmetries.
Fixing all these symmetries we find that there are only two physical degrees of freedom
left. Then we can ask the question whether it is possible to formulate the action for
these physical degrees of freedom that is not based on the principle of covariance of the
action under general diffeomorphism. The construction of such an action was proposed
recently in two very interesting papers [18, 17]. The basic idea presented there was to
perform the conformal traceless decomposition of the gravitational field [6] so that we
have one degrees of freedom corresponding to the scale factor of the metric while we
have five degrees of freedom of the metric that is restricted to have unit determinant.
Then it was shown in [18] that by gauge fixing of the Hamiltonian constraint one can
eliminate the scale factor together with the conjugate momenta. As a result we obtain the
action for five degrees of freedom that is invariant under spatial diffeomorphism wherenow
Hamiltonian is determined by the solving of the Hamiltonian constraint of the General
Relativity for the momentum conjugate to the scale factor. This analysis was further
generalized in a very nice paper in [17] where the starting point was the action for the five
physical degrees of freedom whereit is required that given theory is invariant underspatial
diffeomorphism. In other words we demand that the constraints that are generators of the
spatial diffeomorphism are the first class constraints. We also have to require that these
generators are preserved during the time evolution of the system. Then the requirement
of the closure of the algebra of the Poisson brackets of these constraints together with the
requirement of their time preservation determines the form of the Hamiltonian and the
form of these constraints. When it is presumed that these constraints depend on partial
derivatives of g trough the scalar curvature we find that the original General Relativity
ij
action is reproduced.
The goal of this paper is to formulate HL gravity for the gravitational physical degrees
of freedom only in the similar way as in [18]. To do this we start with another version of
HL gravity that has the correct number of physical degrees of freedom and which is known
as Lagrange multiplier modified HL gravity [7]. This model is based on the formulation of
the HL gravity with reduced symmetry group known as restricted-foliation-preserving Diff
(RFDiff) HLgravity [12,7]. Thisisthetheorythatisinvariant underfollowing symmetries
t′ = t+δt ,δt = const , x′i = xi+ζi(x,t) . (1.3)
Thecharacteristic propertyof Lagrangemultiplier modifiedHL gravity isan absenceof the
Hamiltonian constraint [13] and also presence of the additional constraint which changes
the constraint structure of given theory so that the number of physical degrees of freedom
is the same as in the case of General Relativity. Then in order to separate physical degrees
of freedom of HL gravity we perform conformal traceless decomposition of the gravita-
tional field, following [6, 8]. In this procedurewe introduce new additional scalar field with
additional symmetry so that the number of physical degrees of freedom is the same. Per-
forming Hamiltonian analysis we also identify two second class constraints that, together
with the gauge fixing scaling symmetry allow us to find Hamiltonian for the physical de-
grees of freedom, at least in principle. These physical degrees of freedom are metric with
unit determinant and conjugate traceless momenta so that the number of physical degrees
– 2 –
of freedom is the same as in the case of General Relativity. On the other hand there are
also important differences. Since this theory arises from the theory with the complicated
second class constraints we find that there is a very complicated symplectic structure on
the phase space of the physical degrees of freedom. Secondly, even if we can claim that
these second class constraints can be solved in principle we find that their solutions have
the form of the non-local perturbative expansions. In other words it is hard to see how
such a theory could be useful for some practical computations or even for its path integral
formulation.
However we mean that the analysis performed here suggests very interesting direc-
tion in further research. The starting point would be the general form of the action for
the physical degrees of freedom as was analyzed in [17] where we now presume that the
additional term in the diffeomorphism constraint depends either on higher order of scalar
curvature as for example R Rij or it depends on R non-locally. Then we should proceed
ij ij
as in [17] where we demand that the Poisson brackets of the spatial diffeomorphism con-
straints close on the constraint surface. Then from the requirement of the preservation of
theseconstraintsduringthetimeevolution ofthesystemwecoulddeterminecorresponding
Hamiltonian density. We hope to return to this problem in future.
2. Brief Review of Lagrange Multiplier Modified HL Gravity
We begin this section with the brief review of the Lagrange multiplier modified RFDiff
invariant HL gravity, for more detailed treatment see [7]. RFDiff invariant Hoˇrava-Lifshitz
gravity was introduced in [12], see also [13]. In [7] this action was extended by introducing
Lagrange multiplier term that ensures that the spatial curvature is constant. Explic-
itly,Lagrange multiplier modified RFDiff HL gravity has the form
1
S = dtd3x√h(K˜ ijklK˜ (h)+ [R] ) , (2.1)
κ2 Z ijG kl−V G A
where [R] = R Ω, where Ω is constant, is Lagrange multiplier that transforms as
G − A
scalar
′ ′ ′
(t,x) = (t,x) (2.2)
A A
under (1.3). Further, K˜ introduced in (2.1) is modified extrinsic curvature
ij
1
K˜ = (∂ h N N ) (2.3)
ij t ij i j j i
2 −∇ −∇
that differs from the standard extrinsic curvature by absence of the lapse N(t). Further
the generalized De Witt metric ijkl is defined as
G
1
ijkl = (hikhjl+hilhjk) λhijhkl , (2.4)
G 2 −
where λ is a real constant that in case of General Relativity is equal to one. Finally
(h) is a general function of h and its covariant derivative. The analysis performed in
ij
V
[7] showed that this theory possesses the same number of physical degrees of freedom as
General Relativity. For that reason we mean that this action is a good candidate for the
conformal traceless decomposition of the gravitational field and possible identification of
the physical degrees of freedom of HL gravity.
– 3 –
3. Conformal Traceless Decomposition
The conformal-traceless decomposition of the gravitational field was firstly performed in
3
[5] in its initial value problem . In order to implement conformal-traceless decomposition
we follow [6] and define h and K˜ as
ij ij
1
h = φ4g , K˜ = φ−2A + φ4g τ . (3.1)
ij ij ij ij ij
3
We see that this definition is redundant since the multiple of the fields g ,φ,A ,τ give
ij ij
the same physical metric h and modified extrinsic curvature K˜ . In fact, we see that the
ij ij
decomposition (3.1) is invariant under the conformal transformation
′ 4 ′ −1
g (x,t) = Ω (x,t)g (x,t) , φ(x,t) = Ω (x,t)φ(x,t) ,
ij ij
′ −2 ′
A (x,t) = Ω (x,t)A (x,t), τ (x,t) = τ(x,t) .
ij ij
(3.2)
We also see that (3.1) is invariant under following transformation
1
′ ′ 6
τ (x,t) = τ(x,t)+ζ(x,t) , A (x,t) = A (x,t) ζ(x,t)φ g (x,t) . (3.3)
ij ij − 3 ij
Clearly the gauge fixing of these symmetries we can eliminate τ and φ.
In order to perform the Hamiltonian analysis of the conformal decomposition of the
action (2.1) we firstly rewrite the action (2.1) to its Hamiltonian form. To do this we
introduce the conjugate momenta
δS 1 δS δS
Pij = δ∂ h = κ2√hGijklK˜kl , Pi = δ∂ Ni = 0 , PA = δ∂ ≈ 0 .
t ij t t
A
(3.4)
Then we easily determine corresponding Hamiltonian
H = d3x(∂ h Pij )= d3x( ′ +Ni ′) , (3.5)
t ij T i
Z −L Z H H
where
2
κ
′ = Pij Pkl+√g (h) √h (R) , ′ = 2h Pjk . (3.6)
HT √h Gijkl V − AG Hi − ij∇k
Using the Hamiltonian and the corresponding canonical variables we write the action (2.1)
as
S = dtL = dtd3x(Pij∂ h )= dtd3x(Pij∂ h N ′ Ni ′) . (3.7)
t ij t ij T i
Z Z −H Z − H − H
Then we insert the decomposition (3.1) into the definition of the canonical momenta Pij
1 1
Pij = √g(φ−4˜ijklA + φ2τ ˜ijklg ) , (3.8)
κ2 G kl 3 G kl
3Forreview and extensivelist of references, see [14].
– 4 –
where the metric ˜ijkl is defined as
G
1
˜ijkl = (gikgjl+gilgjk) λgijgkl , ijkl = φ−8˜ijkl . (3.9)
G 2 − G G
Note that ˜ijkl has the inverse
G
1 λ
˜ = (g g +g g ) g g , ˜ = φ8 . (3.10)
ijkl ik jl il jk ij kl ijkl ijkl
G 2 − 3λ 1 G G
−
Using (3.8) and (3.1) we rewrite Pij∂ h into the form
t ij
Pij∂ h = 1 √g˜ijklA + √gφ6(1 3λ)τgij ∂ g +
t ij (cid:18)κ2 G kl 3κ2 − (cid:19) t ji
+ 4 √gφ−1A gkl(1 3λ)+ 4√g(1 3λ)φ5τ ∂ φ .
(cid:18)κ2 kl − κ2 − (cid:19) t
(3.11)
We see that it is natural to identify the expression in the parenthesis with momentum πij
conjugate to g and p conjugate to φ respectively
ij φ
πij = 1 √g˜ijklA + √g(1 3λ)φ6τgij ,
κ2 G kl 3κ2 −
p = 4 √gφ−1A gji(1 3λ)+ 4√g(1 3λ)φ5τ .
φ κ2 ij − κ2 −
(3.12)
Then using (3.12) we obtain following primary constraint
Σ : p φ 4πijg = 0 . (3.13)
D φ ji
−
As we will see below this is the constraint that generates conformal transformation of the
dynamical fields. Further, using(3.12) wefindtherelation between Pij and πij inthe form
Pij = φ−4πij . (3.14)
Then we find that the kinetic term in takes the form
T
H
κ2 κ2φ−6
Pij Pkl = πij ˜ πkl . (3.15)
ijkl ijkl
√h G √g G
As the next step we introduce the decomposition (3.1) into the contribution d3xNi ′.
Hi
Using the relation between Levi-Civita connections evaluated with the metric cRomponents
h and g
ij ij
1
Γk(h) = Γk(g)+2 (∂ φδk +∂ φδk ∂ φgklg ) (3.16)
ij ij φ i j j i − l ij
4
and also if we define n through the relation N = φ n we obtain
i i i
d3xNi ′ = d3xni ′′ ,
i i
Z H Z H
(3.17)
– 5 –
where
′′ = 2g D πjk +4φ−1∂ φg πkl , (3.18)
i ik j i kl
H −
where the covariant derivative D is defined using the Levi-Civita connection Γk(g). Ob-
i ij
′′
serve that with the help of the constraint Σ we can write the constraint as
D Hi
′′ = 2g D πjk +∂ φp 4φ−1∂ φΣ ˆ 4φ−1∂ φΣ (3.19)
i ik j i φ i D i i D
H − − ≡ H −
so that we see that it is natural to identify ˆ as an independent constraint. In fact, we
i
H
will see that the smeared form of this constraint generates the spatial diffeomorphism.
Finally we should proceed to the analysis of the spatial curvature and generally the
whole potential term . Note that this is the function of the covariant derivative, R and
V
R . Using the following formulas
ij
6 2 g 2
R [h] = R [g]+ D φD φ D D φ 2 ijD [gklD φ] g D φgklD φ ,
ij ij φ2 i j − φ i j − φ k l − φ2 ij k l
8
R[h] = φ−4[R[g] gijD D φ] ,
i j
− φ
(3.20)
Then using also the relation between Levi-Civita connections evaluated on h and g we find
that the potential term is generally function of φ and g whose explicit form is not needed
here. As a result we find the action in the form
κ2φ−6
S = dtd3x(πij∂ g +p ∂ φ ni ˆ πij ˜ πkl √gφ6 (φ,h)+
t ij φ t i ijkl
Z − H − √g G − V
8
+ √gφ6 (φ−4R[g] gijD D φ) λΣ ) ,
AG − φ5 i j − D
(3.21)
where we included the primary constraint Σ multiplied by the Lagrange multiplier λ.
D
NowwecanproceedtotheHamiltonian analysis oftheconformaldecomposition ofthe
gravitational fieldgiven bytheaction (3.21). Clearly wehavefollowingprimaryconstraints
πi 0 ,πA 0 , ΣD 0 , (3.22)
≈ ≈ ≈
where πi and πA are momenta conjugate to ni and with following non-zero Poisson
A
brackets
ni(x),πj(y) = δjiδ(x y) , (x),πA(y) = δ(x y) . (3.23)
− {A } −
(cid:8) (cid:9)
Further, the preservation of the primary constraints πi and πA implies following secondary
ones
1
Hˆi ≈ 0 ,Φ1 ≡ κ2√gφ6G ≈ 0 . (3.24)
Now we should analyze the requirement of the preservation of the primary constraint Σ
D
during the time evolution of the system. First of all the explicit calculations give
Σ (x),g (y) = 4g (x)δ(x y) ,
D ij ij
{ } −
– 6 –
Σ (x),πij(y) = 4πij(x)δ(x y) ,
D
− −
(cid:8) (cid:9)
Σ (x),φ(y) = φ(x)δ(x y) ,
D
{ } − −
Σ (x),p (y) = φ(x)δ(x y)
D φ
{ } −
(3.25)
using the canonical Poisson brackets
1
g (x),πkl(y) = (δkδl +δlδk)δ(x y) , φ(x),p (y) = δ(x y) . (3.26)
ij 2 i j i j − { φ } −
n o
It turns out that it is useful to introduce the smeared forms of the constraints ˆ ,Σ
i D
H
T (Ni) = d3xNi ˆ , D(M) = d3xMΣ , (3.27)
S i D
Z H Z
where Ni and M are smooth functions on R3. Then using (3.25) and also
Σ (x),Γk(y) = 2δk∂ δ(x y)+2δk∂ δ(x y) 2gkl(y)∂ δ(x y)g (y)
D ij j yi i yj yl ij
− − − −
n o
(3.28)
we easily find that
′
D(M), (y) = 0 , (3.29)
T
H
(cid:8) (cid:9)
where
κ2φ−6
′ = πij ˜ πkl+√gφ6 (φ,g) . (3.30)
HT √g Gijkl V
To proceed further we use following Poisson brackets
T (Ni),g (x) = Nk∂ g (x) ∂ Nkg (x) g ∂ Nk(x) ,
S ij k ij i kj ik j
− − −
(cid:8)T (Ni),πij(x)(cid:9) = ∂ (Nkπij)(x)+∂ Niπkj(x)+πik∂ Nj(x) ,
S k k k
−
(cid:8) T (Ni),φ(x)(cid:9) = Ni∂ φ(x) ,
S i
−
(cid:8)T (Ni),p (x)(cid:9) = ∂ (Nip )(x)
S φ i φ
−
(cid:8) (cid:9)
(3.31)
and hence it is easy to see that
T (Ni),Σ (x) = Ni∂ Σ (x) ∂ NiΣ (x) (3.32)
S D i D i D
− −
(cid:8) (cid:9)
that together with (3.29) implies that Σ 0 is the first class constraint.
D
≈
Now weproceedto theanalysis ofthepreservation of thesecondary constraints ˆ 0
i
H ≈
and Φ1 0. Note that the total Hamiltonian takes the form
≈
HT = d3x( T′ +λΣD +ni ˆi+γpA+ΓIΦ1) ,
Z H H
(3.33)
where γ is the Lagrange multiplier corresponding to the constraint pA while ΓI is the
Lagrange multiplier corresponding to the constraint Φ1 0.
≈
– 7 –
In case of ˆ we find following Poisson brackets
i
H
∂ ∂
ˆ (x), ˆ (y) = ˆ (x) δ(x y) ˆ (y) δ(x y) (3.34)
Hi Hj Hj ∂xi − −Hi ∂yj −
n o
which implies that the smeared form of the diffeomorphism constraints takes the familiar
form
T (Ni),T (Mj) = T (Nj∂ Mi Mj∂ Ni) . (3.35)
S S S j j
−
(cid:8) (cid:9)
Further using (3.31) we easily find
T (Ni), ′ (x) = ∂ Ni ′ (x) Ni∂ ′ (x) ,
S T i T i T
H − H − H
(cid:8) TS(Ni),Φ1((cid:9)x) = ∂iNiΦ1(x) Ni∂iΦ1(x)
− −
(cid:8) (cid:9)
(3.36)
that implies that ˆ are the first class constraints that are preserved during the time
i
H
evolution of the system. Finally we analyze the time evolution of the constraint Φ1 0 .
≈
Using following formulas
R(x),πij(y) = Rij(x)δ(x y)+DiDjδ(x y) gijD Dkδ(x y) ,
k
− − − − −
(cid:8) (cid:9) 1
Γk(x),πmn(y) = gkp[D δ(δmδn +δnδm)δ(x y)+
ij 4 i j p j p −
n o
+ D (δmδn +δnδm)δ(x y) D (δmδn +δnδm)δ(x y)]
j p i p i p i j i j
− − −
(3.37)
we find that the time derivative of Φ1 is equal to
∂tΦ1 = Φ1,HT
{ } ≈
2
2κ λ
(R πij Rπ)+
− φ4√g ij − 3λ 1
−
2 2 2 2
2κ φ 2κ φ (1 λ)
+ D D [φ−6πkl]+ − D Dk[φ−6π]
k l k
√g √g 3λ 1 −
−
2
κ λ
16 (πij gijπ)D D φ+
− φ5√g − 3λ 1 i j
−
2 2
16φκ 8κ φ2λ 1
+ DiφDj[φ−6πij] − DiφgijDj[φ−6π] Φ2 ,
√g − √g 3λ 1 ≡
−
(3.38)
where Φ2 is an additional constraint that has to be imposed on the system. Following
[19, 20, 21] we include the constraint Φ2 into the definition of the total Hamiltonian that
now has the form
H = d3x( ′ √gφ6 (φ−4R[g] 8φ−5gijD D φ Ω)+λΣ +
T T i j D
Z H − A − −
+ ni ˆi+γpA+ΓIΦ1+ΓIIΦ2) .
H
(3.39)
– 8 –
Now weshouldagain check thestability ofallconstraints. Itis easytoseethattheprimary
constraints together with T (Ni) are preserved while the time evolution of the constraint
S
Φ1 0 is equal to
≈
∂tΦ1 = Φ1,HT d3x ΓII(x) Φ1,Φ2(x)
{ }≈ Z { } ≈
(cid:0) (cid:1)
d3xΓII(x) Φ1,Φ2(x) = 0 .
≈ Z { }
(3.40)
As follows from the explicit form of the constraints Φ1,2 we have
Φ1(x),Φ2(y) = 0 . (3.41)
{ } 6
Then we find that the equation (3.40) gives ΓII = 0. In the same way the requirement of
the preservation of the constraint Φ2 implies
∂tΦ2 dDx( Φ2, T(x) +ΓI(x) Φ2,Φ1(x) )= 0 .
≈ Z { H } { }
(3.42)
Using the fact that Φ2, T(x) = 0 and also the equation (3.41) we see that (3.42) can
{ H } 6
be solved for ΓI. In fact, (3.41) shows that Φ1 and Φ2 are the second class constraints. We
also see from the previous analysis that no additional constraints have to be imposed on
the system.
In order to find the action for the physical degrees of freedom we have to finally fix
the gauge symmetry generated by Σ . To do this we introduce the gauge fixing function
D
Φ = √g 1 . (3.43)
G.F.
−
It is easy to see that there is non-zero Poisson bracket between Φ and Σ so that they
G.F. D
are the second class constraints. In summary we have following collection of the second
class constraints
Φ1 = 0 ,Φ2 = 0 ,ΣD = 0 ,ΦG.F. = 0 . (3.44)
Thegoalistoeliminatesomedegreesoffreedomfromtheseconstraints,atleastinprinciple.
In fact, from Φ1, which is version of Lichnerowitz-York equation [22], we express φ as
1
−1 5
φ= (φR[g] φ Ω) . (3.45)
8∇ −
where −1 is inverse operator to gijD D . We can solve the equation above perturbatively
i j
∇
around some constant φ0. Further, from ΣD we express pφ. Finally, ΦG.F. reduces number
of degrees of freedom in g to be equal to five and from Φ2 we express π as the function of
remaining degrees of freedom. In summary, the physical degrees of freedom of Lagrange
multiplier modified HL gravity are
g , √g = 1 , π˜ij ,g π˜ji = 0 . (3.46)
ij ij
– 9 –