aa r X i v : . [ h e p - t h ] J u l Preprint typeset in JHEP style - HYPER VERSION hep-th/0907.3566
Hoˇrava-Lifshitz f ( R ) Gravity
J. Klusoˇn
Department of Theoretical Physics and AstrophysicsFaculty of Science, Masaryk UniversityKotl´aˇrsk´a 2, 611 37, BrnoCzech RepublicE-mail: [email protected]
Abstract:
This paper is devoted to the construction of new type of f ( R ) theories ofgravity that are based on the principle of detailed balance. We discuss two versions ofthese theories with and without the projectability condition. Keywords:
Hoˇrava-Lifshitz theory. ontents
1. Introduction 12. Non-Linear Scalar Lifshitz Theory 33. Hoˇrava-Lifshitz f ( R ) Theory of Gravity-With Projectability Condition 6
4. Hoˇrava-Lifshitz f ( R ) Theory of Gravity-Without Projectability Condi-tion 12
1. Introduction
Recently in series of very interesting papers P. Hoˇrava suggested new approach for thestudy of membranes and quantum gravity theories known as Hoˇrava-Lifshitz gravities [1,2, 3, 4] . The attractive property of Hoˇrava-Lifshitz gravity is that it is power-countingrenormalizable. The second important property of the Hoˇrava construction is detailedbalance condition . This condition is based on the idea that the potential term in theLagrangian of D + 1 dimensional theory descants from the variation of D dimensionalaction. In fact, this construction is based on the following idea known from the condensedmatter physics [58]: Is it possible to find such a D + 1 dimensional quantum theory suchthat its ground state wave functional reproduces the partition function of D dimensionaltheory? This idea was elaborated in details in [58] and recently in series of papers byP. Hoˇrava with many interesting results. In particular, if we start with known classicaluniversality classes in D dimension we can construct a quantum critical systems in D + 1dimensions.It is very interesting that similar situation naturally occurs in case of topological stringtheory [59, 60], OSV conjecture, topological M-theory, together with non-critical M-theory[61, 62, 63] .As was carefully discussed recently in [11], there are at least four versions of thetheory: with/without the detailed balance condition; and with/without the projectability Hoˇrava’s ideas were elaborated from different points of view in couple of papers, see for example[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 68, 69, 70, 71, 72]. For recent discussion and extensive list of references, see [64]. – 1 –ondition. As we will show bellow the projectability condition means that the lapse functiondepends on time only. It was argued in many papers that the most promising is the versionwithout the detailed balance condition with the projectability condition that has a potentialto be theoretically consistent and cosmologically viable.Even if there are doubts considering the detailed balance condition in general relativitywe feel that it deserves to be studied further. In particular, let us consider followingsituation when we have a D dimensional quantum theory with corresponding partitionfunction Z . Then we consider Hamiltonian of D + 1 dimensional theory and ask thequestion under which condition this Hamiltonian annihilates the vacuum wave functionalof given D + 1 dimensional theory where the norm of this vacuum state coincides with thepartition function Z . We show that there exists an infinite number of such Hamiltoniansthat can be defined as a Taylor series in powers of creation and annihilation operatorswhere the annihilation operator annihilates the vacuum wave functional.We apply this construction to the case of theory of gravity and consider two situations.In the first one we follow the original Hoˇrava’s approach [3] where we start with partitionfunction of D dimensional gravity and demand that there exists quantum gravity in D + 1dimension such that the norm of the ground state functional coincides with the partitionfunction of D dimensional theory. Then we show that we can construct infinite number ofHamiltonians that annihilate this ground state. In other words we find an infinite numberof Hamiltonians that obey the detailed balance conditions. Clearly these Hamiltoniansare well defined at the classical level due to the well known peculiarities that arise innon-linear quantum theories. Even at the classical level these Hamiltonians have manyinteresting properties that should be studied further. For example, for the special form ofthe Hamiltonian that will be specified below we determine corresponding Lagrangian andwe find that the action for this theory is manifestly invariant under spatial diffeomorphism.Then, following [3] we perform an extension of given symmetries that leads to the actionthat is invariant under foliation preserving diffeomorphism . We find new non-linear theoryof gravity that resemble f ( R ) theory of gravity that however is not invariant underfull D + 1 diffeomorphism. It would be certainly interesting to study the cosmologicalimplications of this model exactly in the same way as in case of ordinary f ( R ) theory ofgravity. Certainly we can also consider the more general form of Hamiltonians then theHamiltonian explicitly studied in this paper.In the second case we consider an alternative form of the principle of detailed balance.We consider the situation when the Hamiltonian density of D + 1 dimensional theory is alinear combination of the diffeomorphism and Hamiltonian constraint. Further we assumethat the Hamiltonian constraint has the special property that it annihilates the vacuumwave functional that has the norm equal to the partition function of D dimensional theoryof gravity. Since this vacuum wave functional is manifestly invariant under D dimensionalspatial diffeomorphisms it is annihilated by the generator of diffeomorphism and conse-quently by the Hamiltonian of the theory. We would like to stress that at this moment weonly assume that the Hamiltonian annihilates the vacuum wave functional but we do not For review and extensive list of references, see [65, 73, 74]. – 2 –emand that it should annihilate all states of the theory. On the other hand we will arguethat the Hamiltonian framework implies that the Hamiltonian should annihilate all statesin case of the quantum mechanical formulation of the theory. Explicitly, following thestandard approach we determine an action corresponding to given Hamiltonian. Then wecontinue in the study of this theory and develop the Hamiltonian formalism that followsfrom this action. Since the action contains the fields N ( t, x ) and N i ( t, x ) without timederivatives we find that the absence of corresponding momenta imply the primary con-straints of the theory π N ( x ) ≈ , π i ( x ) ≈
0. The consistency of these constraints with thetime evolution of the system implies the secondary constraints H ( x ) ≈ , H i ( x ) ≈ all quantum states of the theory have tobe annihilated by these constraints as opposite to the original assumption that the statethat should be annihilated by H is the vacuum state only. On the other hand we willargue that these theories suffer from the same problems as the Hoˇrava’s theories withoutprojectability conditions [37]. However using the fact that these theories are constructedas theories that obey the detailed balance conditions it is possible to find the algebra ofconstraints that close however that do not support any physical excitations at all. In otherwords these theories are topological. As the second example of solvable theory we considerthe case of ultralocal theory and we argue the algebra of constraints closes as in standardgravity theory.Let us outline our results. Imposing the detailed balance condition we are able to findnew D + 1 f ( R ) theories of gravity with or without projectability conditions. We wouldlike to stress that these theories should be considered as toy models of gravity theories. Itwould be interesting to study the cosmological implications of these theories in the sameway as in case of f ( R ) theories of gravity with full diffeomorphism invariance.This paper is organized as follows. In next section (2) we present the main idea of ourconstruction on the simple case of collection of D scalar fields in p + 1 dimensions. Thenin section (3) we perform the construction of new D + 1 dimensional theory of gravitythat is invariant under foliation preserving diffeomorphism. In section (4) we suggest analternative way how to impose the condition of the detailed balance in case of D + 1dimensional theory of gravity and we argue that this procedure leads to D + 1 dimensionaltheory without projectability condition.
2. Non-Linear Scalar Lifshitz Theory
In this section we describe the construction of non-linear Lifshitz theory based on the de-tailed balance condition on the simple example of collection of D scalar fields in p dimen-sions. This procedure is based on an idea is that the norm of the ground state functionalof p + 1 dimensional theory coincides with the partition function of any p dimensionaltheory. We should stress that this requirement is pure formal since we do not carry aboutissues whether this partition function is well defined. Very nice discussion of issues thatare related to the construction of wave functionals can be found in paper [66]. Despite ofthis fact we proceed further and we find that we are able to find new interesting class oftheories at least at the classical level. – 3 –et us start in the same way as in [3, 58] and consider the situation when we havea collection of D scalar fields defined on p dimensional Euclidean space with coordinates x = ( x i ) , i = 1 , . . . , p with following action W = 12 Z d p x δ ij ∂ i Φ M ∂ j Φ N g MN , (2.1)where g MN , M, N = 1 , . . . , D is a constant positive definite symmetric matrix. Clearly wecan consider more general form of the action than the one given in (2.1) but in order toexplain the main idea of the constructions we restrict ourselves to the simple action givenabove.As in standard quantum mechanics the fundamental object of this theory is the par-tition function Z Z = Z D Φ( x ) exp[ − W (Φ( x ))] (2.2)that is defined as a path integral on the space of field configurations Φ M ( x ). Then let usassume an existence of p + 1 dimensional quantum field theory with collection of the opera-tors ˆΦ M ( x ) and their conjugate momenta ˆΠ M ( x ) and that obey the canonical commutationrelation [ ˆΦ M ( x ) , ˆΠ N ( y )] = iδ MN δ ( x − y ) . (2.3)Further, we introduce eigenstate of ˆΦ M ( x ) that is the state | Φ( x ) i that obeysˆΦ M ( x ) | Φ( x ) i = Φ M ( x ) | Φ( x ) i . (2.4)In the Schr¨odinger representation any state of given system is represented as the statefunctional Ψ[Φ( x )] and the standard interpretation of quantum mechanics implies thatΨ[Φ( x )]Ψ ∗ [Φ( x )] is a density on the configuration space. Note also that action of theoperator ˆΦ M ( x ) on this state functional corresponds to multiplication with Φ M ( x ). On theother hand the commutation relation (2.3) implies that in the Schr¨odinger representationthe operator ˆΠ M ( x ) is equal to ˆΠ M ( x ) = − i δδ Φ M ( x ) . (2.5)Our goal is to formulate p + 1 dimensional system with the property that the norm of itsground-state functional Ψ [Φ( x )] reproduces the partition function (2.2) h Ψ | Ψ i = Z D Φ( x )Ψ ∗ [Φ( x )]Ψ [Φ( x )] = Z D Φ( x ) exp[ − W (Φ( x ))] . (2.6)Everything that has been done up to this point is well known. However we now makea presumption that the Hamiltonian of p + 1 dimensional theory has the formˆ H ( x ) = κ ∞ X n =0 ˆ c n ( ˆΦ)( ˆ Q † M g MN ˆ Q N ) n − ˆ c ( ˆΦ) ! , (2.7)– 4 –here κ is a coupling constant, ˆ c n ( ˆΦ) are functions that generally depend on the operatorsˆΦ and where ˆ Q M , ˆ Q † M are defined asˆ Q M = i ˆΠ M + 12 δW [ ˆΦ] δ ˆΦ M ( x ) , ˆ Q † M = − i ˆΠ M + 12 δW [ ˆΦ] δ ˆΦ M ( x ) . (2.8)Note that in the Schr¨odinger representation the operators ˆ Q M , ˆ Q † M are equal toˆ Q M = δδ Φ M ( x ) + 12 δW [Φ] δ Φ M ( x ) , ˆ Q † M = − δδ Φ M ( x ) + 12 δW [Φ] δ Φ M ( x ) . (2.9)Let us assume that the vacuum wave functional takes the formΨ [Φ( x )] = exp (cid:18) − W (cid:19) = exp (cid:18) − Z d p x δ ij ∂ i Φ M ( x ) g MN ∂ j Φ N ( x ) (cid:19) . (2.10)Then it is easy that ˆ Q M defined in (2.9) annihilates Ψ ˆ Q M Ψ[Φ( x )] = 0 (2.11)as follows from the fact that i ˆΠ( x )Ψ [Φ] = δδ Φ M ( x ) Ψ [Φ] = − δWδ Φ M ( x ) Ψ [Φ] . (2.12)In other words the vacuum wave functional is annihilated by ˆ Q M and by construction it isan eigenstate of the Hamiltonian with zero energy. Further, the norm of the vacuum wavefunctional coincides with the partition function of p dimensional theory.It is clear that it this way we can define an infinite number of Hamiltonians thatobey the detailed balance condition. In what follows we restrict ourselves to the followingexample of the Hamiltonian densityˆ H = κ s ˆ α ( ˆΦ) + ˆ β ( ˆΦ) (cid:18) ˆΠ M g MN ˆΠ N + 14 (cid:18) δWδ ˆΦ M g MN δWδ ˆΦ N (cid:19)(cid:19) − q ˆ α ( ˆΦ) ! , (2.13)where ˆ α, ˆ β generally depend on ˆΦ and where the square root function in the definition ofthe Hamiltonian is defined as the Taylor polynomial in ( ˆ Q † ˆ Q ) n written explicitly in (2.7).As the next step we determine the Lagrangian from the classical form of the Hamil-tonian density (2.13). Using the Hamiltonian equation ∂ t Φ = { Φ , H } and the form of theHamiltonian density (2.13) we find ∂ t Φ M = (cid:8) Φ M , H (cid:9) = κ β Π N g NM q α + β (Π M g MN Π N + δWδ Φ M g MN δWδ Φ N ) (2.14)so that the Lagrangian density is equal to L = Π M ∂ t Φ M − H == − κ r α (Φ) + β (Φ)4 δWδ Φ M g MN δWδ Φ N s − κ β (Φ) ∂ t Φ M g MN ∂ t Φ N + κ p α (Φ) . (2.15)– 5 –et us now simplify the action further and consider the case when α = 1 , β = const. Then,since the variation of (2.1) is equal to δWδ Φ M ( x ) = − ∂ i ∂ i Φ N ( x ) g NM (2.16)we find that the action of p + 1 dimensional theory takes the form S = − κ Z d p x dt r β ∂ i ∂ i Φ M ) g MN ( ∂ j ∂ j Φ N ) r − κ β ∂ t Φ M g MN ∂ t Φ N . (2.17)If we now expand this action up to quadratic order in fields we find the standard Lifshitzaction (up to trivial rescaling of β ) S = − κ Z dtd p x − Z dtd p x [ κ β
18 ( ∂ i ∂ i Φ M ) g MN ( ∂ j ∂ j Φ N ) − κ β ∂ t Φ M g MN ∂ t Φ N ] . (2.18)In other words for small spatial and time derivatives the Lagrangian (2.15) reduces to theLifshitz scalar theory.
3. Hoˇrava-Lifshitz f ( R ) Theory of Gravity-With Projectability Condition
Let us now turn to the main topic of this paper which is a construction of the Hoˇrava-Lifshitz f ( R ) theories of gravity in D + 1 dimensions. This construction is based onassumption that we have D +1 dimensional quantum theory of gravity that is characterizedby following quantum Hamiltonian densityˆ H = κ p ˆ g ∞ X n =0 ˆ c n (ˆ g ij )( ˆ Q † ij g ˆ G ijkl ˆ Q kl ) n − ˆ c (ˆ g ij ) ! , (3.1)where ˆ Q † ij = − i ˆ π ij + p ˆ g ˆ E ij (ˆ g ij ) , ˆ Q † ij = − i ˆ π ij + p ˆ g ˆ E ij (ˆ g ij ) , (3.2)and where ˆ g = det ˆ g ij and κ is a coupling constant of given theory. Note that the funda-mental operators of quantum theory of gravity are metric components ˆ g ij ( x ) , i = 1 , . . . , D together with their conjugate momenta ˆ π ij ( x ). These operators obey the commutationrelations [ˆ g ij ( x ) , ˆ π kl ( y )] = 12 ( δ ki δ lj + δ li δ kj ) δ ( x − y ) . (3.3)Further, ˆ c n defined in (3.1) are scalar functions that depend on ˆ g ij only. It is clear that inthe Schr¨odinger representation the operators (3.2) take the formˆ Q ij ( x ) = − δδg ij ( x ) + √ g ( x ) E ij ( x ) , ˆ Q † ij ( x ) = δδg ij ( x ) + √ g ( x ) E ij ( x ) . (3.4)– 6 –he next goal is to specify the form of the operators E ij . To do this we assume that thetheory obeys the detailed balance condition so that √ g ( x ) E ij ( x ) = 12 δWδg ij ( x ) , (3.5)where W is an action of D dimensional gravity. As in [3] we construct the vacuum wavefunctional of D + 1 dimensional theory asΨ[ g ( x )] = exp (cid:18) − W (cid:19) , (3.6)where W is the Einstein-Hilbert action in D dimensions W = 12 κ W Z d D x √ gR . (3.7)Generally the action W could also contains additional terms that are functions of metrichowever the explicit form of W will not be important in following discussion.The form of the vacuum wave functional (3.6) implies that it is annihilated by (3.1).Further as a consequence of the detailed balance condition the norm of the functional (3.6)coincides with the partition function of D dimensional Euclidean gravity. In other wordswe have again infinite number of possible Hamiltonians that annihilate the vacuum state(3.6) and that are defined using the principle of detailed balance.In order to find the Lagrangian formulation of this theory we now consider the classicalform of the Hamiltonian density (3.1). In order to simplify the analysis we restrict ourselvesto the following explicit form of the Hamiltonian density H = κ √ g (cid:18)r β ( − iπ ij + √ gE ij ) 1 g G ijkl ( iπ kl + √ gE kl ) − (cid:19) , (3.8)where G ijkl denotes the inverse of the De Witt metric G ijkl = 12 ( g ik g jl + g il g jk ) − ˜ λg ij g kl (3.9)with ˜ λ = λDλ − . The ”metric on the space of metric”, G ijkl is defined as G ijkl = 12 ( g ik g jl + g il g jk − λg ij g kl ) (3.10)with λ an arbitrary real constant. Note that (3.9) together with (3.10) obey the relation G ijmn G mnkl = 12 ( δ ki δ lj + δ li δ kj ) . (3.11) Note that we use the terminology introduced in [3] and that we review there. In case of relativistictheory, the full diffeomorphism invariance fixes the value of λ uniquely to equal λ = 1. In this case theobject G ijkl is known as the ”De Witt metric”. We use this terminology to more general case when λ is notnecessarily equal to one. – 7 –he form of the Hamiltonian density (3.8) implies following time derivative of g ij ∂ t g ij = { g ij , H } = κ β G ijkl π kl √ g q β ( − iπ ij + √ gE ij ) g G ijkl ( iπ kl + √ gE kl ) . (3.12)With the help of this result we can express π ij as a function of g ij and ∂ t g ij . Then weeasily find the corresponding Lagrangian density in the form L = ∂ t g ij π ij − H = − κ √ g (cid:18)q βE ij G ijkl E kl r − κ β ∂ t g ij G ijkl ∂ t g kl − (cid:19) . (3.13)By construction the action S = Z d D x L , (3.14)where L is given in (3.13) is invariant under the global time translation t ′ = t + δt , δt = constand under the spatial diffeomorphism x ′ i = x i ( x ) . (3.15)This follows from the fact that we presumed that the functional W is invariant under thespatial diffeomorphism under which the metric g ij and tensor E ij transform as g ′ ij ( x ′ ) = g kl ( x ) (cid:0) D − (cid:1) ki (cid:0) D − (cid:1) lj ,E ′ ij ( x ′ ) = E kl ( x ) D ik D jl , (3.16)where D ij = ∂x ′ i ∂x j , D ij (cid:0) D − (cid:1) jk = δ ik . (3.17)Using the transformation property of g ij we find that the metric G ijkl transforms as G ′ ijkl ( x ′ ) = G i ′ j ′ k ′ l ′ ( x ) (cid:0) D − (cid:1) i ′ i (cid:0) D − (cid:1) j ′ j (cid:0) D − (cid:1) k ′ k (cid:0) D − (cid:1) l ′ l (3.18)and the invariance of the action under the spatial diffeomorphism (3.15) is obvious. We argued that the action formulated above is invariant under D dimensional spatial dif-feomorphism . As in [2, 3] we extend these symmetries to the diffeomorphisms that respectthe preferred codimension-one foliation F of the theory by the slices of fixed time. By def-inition such a foliation-preserving diffeomorphism consists a space-time dependent spatialdiffeomorphisms as well as time-dependent time reparameterization. These symmetries arenow generated by infinitesimal transformations δx i ≡ x ′ i − x i = ζ i ( t, x ) , δt ≡ t ′ − t = f ( t ) . (3.19)– 8 –t was shown in [3] that the metric transform under (3.19) as g ′ ij ( t ′ , x ′ ) = g ij ( t, x ) − g il ( t, x ) ∂ j ζ l ( t, x ) − ∂ i ζ k ( t, x ) g kj ( t, x ) . (3.20)The original action (3.14) is not invariant under (3.19). On the other hand it was shown in[3] that in order to find an action that is invariant under (3.19) it is necessary to introducenew fields N i ( t, x ) , N ( t ) that transform under (3.19) as N ′ i ( t ′ , x ′ ) = N i ( t, x ) − N i ( t, x ) ˙ f ( t ) − N j ( t, x ) ∂ i ζ j ( t, x ) − g ij ( t, x ) ˙ ζ j ( t, x ) ,N ′ ( t ′ ) = N ( t ) − N ( t ) ˙ f ( t ) . (3.21)As the next step we have to replace volume element dtd D x √ g with dtd D x N √ g and thetime derivative of g ij with ∂ t g ij ⇒ K ij , (3.22)where K ij is defined as K ij = 12 N ( ∂ t g ij − ∇ i N j − ∇ j N i ) , (3.23)and where ∇ i is D dimensional covariant derivative constructed from the metric compo-nents g ij . It can be shown that (3.23) transform covariantly under (3.19) K ′ ij ( t ′ , x ′ ) = K ij ( t, x ) − K ik ( t, x ) ∂ j ζ k ( t, x ) − ∂ i ζ k ( t, x ) K kj ( t, x ) . (3.24)Performing these substitutions in (3.14) we find the gravity action invariant under thefoliation preserving diffeomorphism in the form S = − κ Z dtd D x √ gN (cid:18)q βE ij G ijkl E kl r − κ β ( K ij K ij − λK ) − (cid:19) . (3.25)Note also that linearized form of the action (3.25) takes the form S = 12 Z dtd D x √ gN (cid:18) κ β ( K ij K ij − λK ) − κ βE ij G ijkl E kl (cid:19) (3.26)that after trivial rescaling of parameter β resembles the Hoˇrava’s form of the gravity theory.For that reason we can consider the action (3.25) as the f ( R )-like version of the Horaˇrava-Lifshitz gravity.In the next subsection we develop the Hamiltonian formalism of given theory.– 9 – .2 Hamiltonian Formalism The dynamical variables of the theory are N i ( x ) , π i ( x ) , N, π N together with g ij ( x ) , π ij ( x )with corresponding non-zero Poisson brackets n g ij ( x ) , π kl ( y ) o = 12 ( δ ki δ lj + δ li δ kj ) δ ( x − y ) , (cid:8) N i ( x ) , π j ( y ) (cid:9) = δ ij δ ( x − y ) , (cid:8) N, π N (cid:9) = 1 . (3.27)Note that N ( t ) and π N ( t ) are homogeneous functions of time. In other words they obeyprojectability condition which has an important consequence for the consistency of theHoˇrava-Lifshitz theory [11]. Further, as follows from the form of the action (3.25) themomenta π ij conjugate to g ij can be expressed as function of g ij and ∂ t g ij from the relation π ij ( x ) = δSδ∂ t g ij ( x ) = 2 κ β √ g G ijkl K kl q − βκ K ij G ijkl K kl q βE ij G ijkl E kl . (3.28)On the other hand since the time derivative of N i and N do not appear in the action (3.25)we find that the momenta π i and π N form the primary constraints of the theory π i ( t, x ) ≈ , π N ( t ) ≈ . (3.29)Finally the standard definition of the Hamiltonian density gives H = ∂ t g ij π ij − L == κ √ gN (cid:18)r g π ij G ijkl π kl + βE ij G ijkl E kl − (cid:19) ++ ( ∇ i N j + ∇ j N i ) π ij . (3.30)As a consequence we find that the Hamiltonian is equal to H = Z d D x H = Z d D x ( N ( t ) H ( x , t ) + N i ( t, x ) H i ( x , t )) , H = κ √ g (cid:18)r g π ij G ijkl π kl + E ij G ijkl E kl − (cid:19) , H i = − ∇ j π ij , (3.31)where we ignore boundary terms.The primary constraints π i ( x ) ≈ , π N ( t ) ≈ ∂ t π i ( x ) = (cid:8) π i ( x ) , H (cid:9) = −H i ( x ) ≈ , ∂ t π N ( t ) = { N ( t ) , H } = − Z d D x H ( x ) ≈ . (3.32)Since the right side of the equations above have to vanish on constraint surface we findthat the consistency of the primary constraints generate the secondary ones H i ( x ) ≈ , T T ≡ Z d D x H ( x ) ≈ . (3.33)– 10 –t is convenient to introduce the smeared form of the diffeomorphism constant T S definedas T S ( ζ ) = Z d D x ζ i ( x ) H i ( x ) . (3.34)The next goal is to calculate the Poisson bracket of constraints T T and T S . Trivially wehave that { T T , T T } = 0 . (3.35)Now we calculate the Poisson brackets between T S ( ζ ) and T { T S ( ζ ) , T T } = − Z d D x ( ζ k ∂ k H − ∂ k ( H ) ζ k ) = 0 , (3.36)where we used the Poisson bracket between T S ( ζ ) and H { T S ( ζ ) , H } = − ∂ k ζ k H − ζ k ∂ k H . (3.37)Finally we calculate the Poisson bracket { T S ( ζ ) , T S ( ξ ) } = T S ( ζ i ∂ i ξ − ξ i ∂ i ζ ) . (3.38)In summary we find that the algebra of constraints for generalized Hoˇrava-Lifshitz theorythat respects the projectability condition takes very simple form { T T , T T } = 0 , { T S ( ζ ) , T T } = 0 , { T S ( ζ ) , T S ( ξ ) } = T S ( ζ i ∂ i ξ − ξ i ∂ i ζ ) . (3.39)The fact that the algebra of constraints is closed for any theory of gravity that obeysthe projectability condition is very attractive. This result in contrast with the situation ofgravity without the projectability condition when the algebra is not closed and the structureconstants of the theory depend on phase space variables. On the other hand there are stillmany unsolved problems and issues considering Hoˇrava-Lifshitz gravity theories as wasreviewed carefully in [11, 16] so that these results should be taken with great care. For more detailed calculation, see (4.16). – 11 – . Hoˇrava-Lifshitz f ( R ) Theory of Gravity-Without Projectability Con-dition
In this section we address the question of the formulation of the local form of the conditionof detailed balance . We again start with the assumption that one can define the vacuumwave functional of D + 1-dimensional quantum theory and that this functional has the formas in (3.6). Now we demand that this vacuum wave functional is annihilated byˆ H = Z d D x (cid:16) N ( t, x ) ˆ H ( t, x ) + N i ( t, x ) ˆ H i ( t, x ) (cid:17) , (4.1)where ˆ H i is the generator of spatial diffeomorphismˆ H i ( x ) = − ∇ j ˆ π ji ( x ) , (4.2)and where we assume that ˆ H can be written asˆ H ( x ) = κ p ˆ g ∞ X n =0 ˆ c n (ˆ g ij )( ˆ Q † ij g ˆ G ijkl ˆ Q kl ) n − ˆ c (ˆ g ij ) ! , (4.3)where ˆ Q ij , ˆ Q † ij were defined in (3.2) and the functional form of ˆ E ij follows from (3.5).Then it is obvious that the local constraint ˆ H ( x ) annihilates the vacuum wave functional(3.6). Since W is invariant under spatial diffeomorphism by construction we find that thevacuum wave functional Ψ is annihilated by ˆ H (4.1) as well. We should again stress theimportant fact that (4.1) contains the lapse function that depends on x as well. Note thatwe only demand that this Hamiltonian annihilates the vacuum state functional while itsaction on other states of the theory is not specified. This is different from the standardconstraint of general relativity where the Dirac analysis implies that all wave functionalsshould be annihilated by Hamiltonian and diffeomorphism constraints. On the other handwe will see below that the correct Hamiltonian treatment of the theory specified by theHamiltonian above will lead to the requirement that all states should be annihilated by(4.1).As usual we are interested in the Lagrangian formulation of given theory. In order tofind it we consider the classical form of the Hamiltonian (4.1) and we also restrict ourselvesto the following form of the Hamiltonian density H H = κ √ g (cid:18)r β ( − iπ ij + √ gE ij ) 1 g G ijkl ( iπ kl + √ g E kl ) − (cid:19) . (4.4)Then using (4.1) and (4.4) we find that the time derivative of g ij is equal to ∂ t g ij = { g ij , H } = κ N β √ g G ijkl π kl q βg π ij G ijkl π kl + βE ij G ijkl E kl + ∇ j N i + ∇ i N j , (4.5)– 12 –here we used the canonical Poisson brackets n g ij ( x ) , π kl ( y ) o = 12 ( δ ki δ lj + δ li δ kj ) δ ( x − y ) (4.6)and the fact that (cid:26) g ij ( x ) , Z d D y N k ( y ) H k ( y ) (cid:27) = − Z d D y N k ( y ) ∇ l ( n g ij ( x ) , π kl ( y ) o ) == ∇ j N i ( x ) + ∇ i N j ( x ) . (4.7)The equation (4.5) implies that it is natural to introduce the tensor K ij = N ( ∂ t g ij −∇ j N i − ∇ i N j ) so that (4.5) can be written as2 K ij = κ β √ g G ijkl π kl q βg π ij G ijkl π kl + βE ij G ijkl E kl . (4.8)Clearly using this relation we can express π ij as a function of g ij , K ij . Then after somealgebra we find the Lagrangian in the form L = Z d D x ( ∂ t g ij π ij − N H − N i H i ) == − κ Z d D x √ gN (cid:18)q βE ij G ijkl E kl r − κ β K ij G ijkl K kl − (cid:19) . (4.9)We see that this Lagrangian takes completely the same form as the Lagrangian given in(3.25). However it is crucial that in the new formulation the field N depends on x and t as well. In other words we derived the Hoˇrava-Lifshitz f ( R ) gravity theory withoutprojectability condition. We see that the Lagrangian density (4.9) depends on N ( t, x ) and N i ( t, x ) that can beinterpreted as additional fields in the theory. Then when we proceed to the Hamiltonianformalism we find that the phase space of the theory is spanned by N, N i with conjugatemomenta π N , π i and metric components g ij with conjugate momenta π ij . The fact that theLagrangian (4.9) does not contain time derivatives of N and N i implies that the momenta π i ( x ) , π N ( x ) vanish and form the primary constraints of the theory. Finally the standardanalysis of constraints system implies that the Hamiltonian (4.1) with H given in (4.4) isa sum of the local constraints H ( x ) ≈ , H i ( x ) ≈ . (4.10)The quantum mechanical analogue of these constraints is the requirement that all wavefunctionals should be annihilated by them. Observe that this is more stronger requirement– 13 –hen the formulation of the local balance condition given in the first paragraph of thissection. In summary, the consistency of the theory defined by (4.9) implies that at theclassical level the Hamiltonian (4.1) should be sum of local constraints. The quantummechanical formulation is that all wave functionals should be annihilated by the quantumHamiltonian (4.1) again with ˆ H given in (4.4).Now we start to study the algebra of constraints H i , H when we presume the mostgeneral form of the constraint H H = κ √ g ∞ X n =0 c n ( g ij ) (cid:18) Q † ij g G ijkl Q kl (cid:19) n − c ( g ij ) ! == κ √ g ∞ X n =1 c n ( g ij ) (cid:18) Q † ij g G ijkl Q kl (cid:19) n . (4.11)If we introduce the smeared form of the diffeomorphism constraint T S ( ζ ) = R d D x ζ i ( x ) H i ( x )we can easily determine Poisson brackets { T S ( ζ ) , g ij } = − ζ k ∂ k g ij − g jk ∂ i ζ k − g ik ∂ j ζ k , (cid:8) T ( ζ ) , π ij (cid:9) = − ∂ k ( π ij ζ k ) + π jk ∂ k ζ i + π ik ∂ k ζ j , { T S ( ζ ) , √ g } = − ζ k ∂ k √ g − ∂ k ζ k √ g , (cid:26) T S ( ζ ) , δWδg ij (cid:27) = − ∂ k (cid:18) ζ k δWδg ij (cid:19) + 12 δWδg ik ∂ j ζ k + ∂ i ζ k δWδg kj . (4.12)Then it is easy to find that (cid:8) T S ( ζ ) , Q ij (cid:9) = − ∂ k ( Q ij ζ k ) + ∂ k ζ i Q kj + Q ik ∂ k ζ j , n T S ( ζ ) , Q † ij o = − ∂ k (cid:16) ζ k Q † ij (cid:17) + ∂ k ζ i Q † kj + Q † ik ∂ k ζ j . (4.13)For further purposes we also determine following Poisson bracket (cid:26) T S ( ζ ) , g G ijkl (cid:27) = (2 ∂ k ζ k ( g ) + ζ k ∂ k ( g )) 1 g G ijkl −− g ( ∂ p G ijkl ζ p + ∂ i ζ p G pjkl + ∂ j ζ p G ipkl + G ijpl ∂ k ζ p + G ijkp ∂ j ζ p ) . (4.14)Then it is easy to see (cid:26) T S ( ζ ) , Q † ij g G ijkl Q kl (cid:27) = − ∂ m (cid:18) Q † ij g G ijkl Q kl (cid:19) ζ m . (4.15)– 14 –sing this result and also the third equation in (4.12) we find { T S ( ζ ) , H } = κ [ − ζ k ∂ k √ g − ∂ k ζ k √ g ] ∞ X n =1 c n (cid:18) Q † ij g G ijkl Q kl (cid:19) n −− κ √ g ∞ X n =1 c n ∂ m (cid:18) Q † ij g G ijkl Q kl (cid:19) ζ m (cid:18) Q † ij g G ijkl Q kl (cid:19) n − == − ∂ k ζ k H − ζ k ∂ k H (4.16)and when we introduce the smeared form of the constraint H T T ( f ) = Z d D x f ( x ) H ( t, x ) (4.17)we obtain { T S ( ζ ) , T T ( f ) } = − Z d D x f ( x )( ∂ k ζ k H ( x ) + ζ k ∂ k H ( x )) == Z d D x ∂ k f ( x ) ζ k H = T T ( ∂ k f ζ k ) . (4.18)Finally the Poisson brackets of the diffeomorphism constraints is equal to { T S ( ζ ) , T S ( ξ ) } = T S ( ζ i ∂ i ξ − ξ i ∂ i ζ ) . (4.19)Now we come to the analysis of the most intricate Poisson bracket { T T ( f ) , T T ( ζ ) } . Notethat the previous Poisson brackets were valid for any form of the constraint H . On theother hand we can certainly find an equivalent constraint using following observation. TheHamiltonian constraint has the form H = f ( Q † ij G ijkl Q kl ). Then instead imposing theconstraint H ≈ √ gQ † ij G ijkl Q kl ≈
0. This fact simplifiesthe analysis considerably however it is still very intricate as was shown for example in [37]where the analysis of the constraint algebra of 3+ 1 dimensional Hoˇrava-Lifshitz theory wasperformed with the result that the Poisson bracket of the constraint √ gQ † ij G ijkl Q kl ≈ H ≈
0. Thisidea was suggested in the original Hoˇrava work [3]. Explicitly, the form of the constraint H ( x ) ≈ H ( x ) ≈ Q ij ( x ) ≈
0. In other words we propose following alternative set of constraints of f ( R )Hoˇrava-Lifshitz gravity H i ( x ) ≈ , Q ij ( x ) ≈ T S ( ζ ) = Z d D x ζ i ( x ) H i ( x ) , Q (Λ) = Z d D x Λ ij ( x ) Q ij ( x ) . (4.21)– 15 –et us now show that this set of constraints forms the closed algebra. Since n Q ij ( x ) , Q kl ( y ) o = − i δδg ij ( x ) δWδg kl ( y ) + i δδg kl ( y ) δWδg ij ( x ) = 0 (4.22)we easily find that { Q (Λ) , Q (Γ) } = 0 . (4.23)Further, using (4.13) we find { T S ( ζ ) , Q (Λ) } = Z d D x ( ∂ k Λ ij Q ij ζ k + ∂ k ζ i Q kj + Q ik ∂ k ζ j ) == Q ( ∂ k Λ ij ζ k + ∂ i ζ k Λ kj + Λ ik ∂ j ζ k ) . (4.24)These Poisson brackets together with (4.19) imply that the algebra of the constraints (4.21)is closed. On the other hand as was stressed originally in [3] this set of constraints is cer-tainly too strong and it turns out that the resulting theory is topological without any localexcitations. This conclusion however suggests that the Hoˇrava-Lifshitz theory of grav-ity without projectability condition has natural physical interpretation as the topologicaltheory of gravity. In this section we present an example of the Hoˇrava-Lifshitz f ( R ) gravity that has closedalgebra of constraints. Using terminology introduced in [2] we call this theory as ultralocalHoˇrava-Lifshitz f ( R ) gravity .The simplest example of the ultralocal theory is characterized by condition that E ij = 0 . (4.25)Since in this case Q ij = − Q † ij we find n Q † ij ( x ) , Q kl ( y ) o = 0 , (cid:26) Q ij ( x ) , g G klmn ( y ) (cid:27) = − (cid:26) Q † ij ( x ) , g G klmn ( y ) (cid:27) (4.26)and consequently (cid:26) Q † ij g G ijkl Q kl ( x ) , Q † mn g G mnpq Q pq ( y ) (cid:27) = 0 . (4.27)Then it is easy to determine the Poisson brackets of the constraints H ≈ {H ( x ) , H ( y ) } = Z d x ′ d y ′ δ H ( x ) δ ( Q † ij G ijkl Q kl )( x ′ ) n ( Q † ij G ijkl Q kl )( x ′ ) , ( Q † ij G ijkl Q kl )( y ′ ) o × – 16 – δ H ( y ) δ ( Q † ij G ijkl Q kl )( y ′ ) = 0 . (4.28)Let us now consider the second example of ultralocal theory when W has the form W = Λ √ g . (4.29)In this case we easily find Q † ij = − iπ ij + 14 Λ g ij √ g , Q ij = iπ ij + 14 Λ g ij √ g . (4.30)Now the Poisson brackets between Q † ij and Q kl is non-zero n Q † ij ( x ) , Q kl ( y ) o = − i Λ4 ( g ik g jl + g il g kl ) √ gδ ( x − y ) . (4.31)It is important that this Poisson bracket is proportional to δ ( x − y ) and does not containderivative of delta function. Then with the help of this result and the second equation in(4.26) we again find that (cid:26) Q † ij g G ijkl Q kl ( x ) , Q † ij g G ijkl Q kl ( y ) (cid:27) = 0 (4.32)and as follows from (4.28) the Poisson brackets of the Hamiltonian constraints vanish.In summary, the ultralocal f ( R ) Hoˇrava-Lifshitz gravity has the same nice propertyas the ultralocal theory of gravity [67]. Acknowledgements:
This work was supported by the Czech Ministry of Educationunder Contract No. MSM 0021622409. – 17 – eferences [1] P. Horava, “Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point,”
Phys. Rev. Lett. (2009) 161301 [arXiv:0902.3657 [hep-th]].[2] P. Horava, “Quantum Gravity at a Lifshitz Point,”
Phys. Rev. D (2009) 084008[arXiv:0901.3775 [hep-th]].[3] P. Horava, “Membranes at Quantum Criticality,” JHEP (2009) 020 [arXiv:0812.4287[hep-th]].[4] P. Horava, “Quantum Criticality and Yang-Mills Gauge Theory,” arXiv:0811.2217 [hep-th].[5] C. Appignani, R. Casadio and S. Shankaranarayanan, “The Cosmological Constant andHorava-Lifshitz Gravity,” arXiv:0907.3121 [hep-th].[6] I. Adam, I. V. Melnikov and S. Theisen, “A Non-Relativistic Weyl Anomaly,” arXiv:0907.2156 [hep-th].[7] A. Wang and R. Maartens, “Linear perturbations of cosmological models in theHorava-Lifshitz theory of gravity without detailed balance,” arXiv:0907.1748 [hep-th].[8] C. Bogdanos and E. N. Saridakis, “Perturbative instabilities in Horava gravity,” arXiv:0907.1636 [hep-th].[9] A. Kobakhidze, “On the infrared limit of Horava’s gravity with the global Hamiltonianconstraint,” arXiv:0906.5401 [hep-th].[10] J. J. Peng and S. Q. Wu, “Hawking Radiation of Black Holes in Infrared ModifiedHoˇrava-Lifshitz Gravity,” arXiv:0906.5121 [hep-th].[11] S. Mukohyama, “Caustic avoidance in Horava-Lifshitz gravity,” arXiv:0906.5069 [hep-th].[12] A. Castillo and A. Larranaga, “Entropy for Black Holes in the Deformed Horava-LifshitzGravity,” arXiv:0906.4380 [gr-qc].[13] R. Iengo, J. G. Russo and M. Serone, “Renormalization group in Lifshitz-type theories,” arXiv:0906.3477 [hep-th].[14] Y. F. Cai and X. Zhang, “Primordial perturbation with a modified dispersion relation,” arXiv:0906.3341 [astro-ph.CO].[15] S. R. Das and G. Murthy, “ CP N − Models at a Lifshitz Point,” arXiv:0906.3261 [hep-th].[16] D. Blas, O. Pujolas and S. Sibiryakov, “On the Extra Mode and Inconsistency of HoravaGravity,” arXiv:0906.3046 [hep-th].[17] Y. F. Cai and E. N. Saridakis, “Non-singular cosmology in a model of non-relativisticgravity,” arXiv:0906.1789 [hep-th].[18] F. W. Shu and Y. S. Wu, “Stochastic Quantization of the Hoˇrava Gravity,” arXiv:0906.1645[hep-th].[19] A. Ghodsi and E. Hatefi, “Extremal rotating solutions in Horava Gravity,” arXiv:0906.1237[hep-th].[20] C. Germani, A. Kehagias and K. Sfetsos, “Relativistic Quantum Gravity at a Lifshitz Point,” arXiv:0906.1201 [hep-th]. – 18 –
21] Y. S. Myung, “Propagations of massive graviton in the deformed Hoˇrava-Lifshitz gravity,” arXiv:0906.0848 [hep-th].[22] M. Botta-Cantcheff, N. Grandi and M. Sturla, “Wormhole solutions to Horava gravity,” arXiv:0906.0582 [hep-th].[23] M. i. Park, “The Black Hole and Cosmological Solutions in IR modified Horava Gravity,” arXiv:0905.4480 [hep-th].[24] M. Sakamoto, “Strong Coupling Quantum Einstein Gravity at a z=2 Lifshitz Point,” arXiv:0905.4326 [hep-th].[25] S. Nojiri and S. D. Odintsov, “Covariant Horava-like renormalizable gravity and its FRWcosmology,” arXiv:0905.4213 [hep-th].[26] A. A. Kocharyan, “Is nonrelativistic gravity possible?,” arXiv:0905.4204 [hep-th].[27] A. Wang and Y. Wu, “Thermodynamics and classification of cosmological models in theHorava-Lifshitz theory of gravity,” arXiv:0905.4117 [hep-th].[28] G. Calcagni, “Detailed balance in Horava-Lifshitz gravity,” arXiv:0905.3740 [hep-th].[29] S. Mukohyama, “Dark matter as integration constant in Horava-Lifshitz gravity,” arXiv:0905.3563 [hep-th].[30] M. Minamitsuji, “Classification of cosmology with arbitrary matter in the Hoˇrava-Lifshitztheory,” arXiv:0905.3892 [astro-ph.CO].[31] X. Gao, Y. Wang, R. Brandenberger and A. Riotto, “Cosmological Perturbations inHoˇrava-Lifshitz Gravity,” arXiv:0905.3821 [hep-th].[32] E. N. Saridakis, “Horava-Lifshitz Dark Energy,” arXiv:0905.3532 [hep-th].[33] Y. W. Kim, H. W. Lee and Y. S. Myung, “Nonpropagation of scalar in the deformedHoˇrava-Lifshitz gravity,” arXiv:0905.3423 [hep-th].[34] G. Bertoldi, B. A. Burrington and A. Peet, “Black Holes in asymptotically Lifshitz spacetimeswith arbitrary critical exponent,” arXiv:0905.3183 [hep-th].[35] A. Dhar, G. Mandal and S. R. Wadia, “Asymptotically free four-fermi theory in 4 dimensionsat the z=3 Lifshitz-like fixed point,” arXiv:0905.2928 [hep-th].[36] T. P. Sotiriou, M. Visser and S. Weinfurtner, “Quantum gravity without Lorentz invariance,” arXiv:0905.2798 [hep-th].[37] M. Li and Y. Pang, “A Trouble with Hoˇrava-Lifshitz Gravity,” arXiv:0905.2751 [hep-th].[38] D. W. Pang, “A Note on Black Holes in Asymptotically Lifshitz Spacetime,” arXiv:0905.2678[hep-th].[39] C. Charmousis, G. Niz, A. Padilla and P. M. Saffin, “Strong coupling in Horava gravity,” arXiv:0905.2579 [hep-th].[40] B. Chen, S. Pi and J. Z. Tang, “Scale Invariant Power Spectrum in Hoˇrava-LifshitzCosmology without Matter,” arXiv:0905.2300 [hep-th].[41] R. A. Konoplya, “Towards constraining of the Horava-Lifshitz gravities,” arXiv:0905.1523[hep-th].[42] J. Kluson, “Stable and Unstable D-Branes at Criticality,” arXiv:0905.1483 [hep-th]. – 19 –
43] S. Chen and J. Jing, “Quasinormal modes of a black hole in the deformed Hˇorava-Lifshitzgravity,” arXiv:0905.1409 [gr-qc].[44] D. Orlando and S. Reffert, “On the Renormalizability of Horava-Lifshitz-type Gravities,” arXiv:0905.0301 [hep-th].[45] R. G. Cai, B. Hu and H. B. Zhang, “Dynamical Scalar Degree of Freedom in Horava-LifshitzGravity,” arXiv:0905.0255 [hep-th].[46] T. P. Sotiriou, M. Visser and S. Weinfurtner, “Phenomenologically viable Lorentz-violatingquantum gravity,” arXiv:0904.4464 [hep-th].[47] X. Gao, “Cosmological Perturbations and Non-Gaussianities in Hoˇrava-Lifshitz Gravity,” arXiv:0904.4187 [hep-th].[48] R. G. Cai, Y. Liu and Y. W. Sun, “On the z=4 Horava-Lifshitz Gravity,”
JHEP (2009)010 [arXiv:0904.4104 [hep-th]].[49] R. G. Cai, L. M. Cao and N. Ohta, “Topological Black Holes in Horava-Lifshitz Gravity,” arXiv:0904.3670 [hep-th].[50] H. Nastase, “On IR solutions in Horava gravity theories,” arXiv:0904.3604 [hep-th].[51] R. Brandenberger, “Matter Bounce in Horava-Lifshitz Cosmology,” arXiv:0904.2835 [hep-th].[52] S. Mukohyama, “Scale-invariant cosmological perturbations from Horava-Lifshitz gravitywithout inflation,”
JCAP (2009) 001 [arXiv:0904.2190 [hep-th]].[53] H. Lu, J. Mei and C. N. Pope, “Solutions to Horava Gravity,” arXiv:0904.1595 [hep-th].[54] J. Kluson, “Branes at Quantum Criticality,” arXiv:0904.1343 [hep-th].[55] E. Kiritsis and G. Kofinas, “Horava-Lifshitz Cosmology,” arXiv:0904.1334 [hep-th].[56] G. Calcagni, “Cosmology of the Lifshitz universe,” arXiv:0904.0829 [hep-th].[57] M. Visser, “Lorentz symmetry breaking as a quantum field theory regulator,” arXiv:0902.0590[hep-th].[58] E. Ardonne, P. Fendley and E. Fradkin, “Topological order and conformal quantum criticalpoints,”
Annals Phys. (2004) 493 [arXiv:cond-mat/0311466].[59] E. Witten, “Quantum background independence in string theory,” arXiv:hep-th/9306122.[60] M. Gunaydin, A. Neitzke and B. Pioline, “Topological wave functions and heat equations,”
JHEP (2006) 070 [arXiv:hep-th/0607200].[61] R. Dijkgraaf, S. Gukov, A. Neitzke and C. Vafa, “Topological M-theory as unification of formtheories of gravity,”
Adv. Theor. Math. Phys. , 603 (2005) [arXiv:hep-th/0411073].[62] P. Horava and C. A. Keeler, “Noncritical M-theory in 2+1 dimensions as a nonrelativisticFermi liquid,” JHEP , 059 (2007) [arXiv:hep-th/0508024].[63] P. Horava and C. A. Keeler, “Thermodynamics of noncritical M-theory and the topologicalA-model,”
Nucl. Phys. B , 1 (2006) [arXiv:hep-th/0512325].[64] R. Dijkgraaf, D. Orlando and S. Reffert, “Relating Field Theories via StochasticQuantization,” arXiv:0903.0732 [hep-th].[65] V. Faraoni, ”f(R) gravity: successes and challenges,” arXiv:0810.2602 [gr-qc]. – 20 –
66] E. Witten, “A note on the Chern-Simons and Kodama wavefunctions,” arXiv:gr-qc/0306083.[67] C. J. Isham, “Some Quantum Field Theory Aspects Of The Superspace Quantization OfGeneral Relativity,”
Proc. Roy. Soc. Lond. A (1976) 209[68] E. O. Colgain and H. Yavartanoo, “Dyonic solution of Horava-Lifshitz Gravity,” arXiv:0904.4357 [hep-th].[69] J. Chen and Y. Wang, “Timelike Geodesic Motion in Horava-Lifshitz Spacetime,” arXiv:0905.2786 [gr-qc].[70] R. G. Cai, L. M. Cao and N. Ohta, “Thermodynamics of Black Holes in Horava-LifshitzGravity,” arXiv:0905.0751 [hep-th].[71] T. Takahashi and J. Soda, “Chiral Primordial Gravitational Waves from a Lifshitz Point,” arXiv:0904.0554 [hep-th].[72] Y. S. Piao, “Primordial Perturbation in Horava-Lifshitz Cosmology,” arXiv:0904.4117[hep-th].[73] S. Nojiri and S. D. Odintsov, “Introduction to modified gravity and gravitational alternativefor dark energy,” eConf
C0602061 , 06 (2006) [Int. J. Geom. Meth. Mod. Phys. , 115(2007)] [arXiv:hep-th/0601213].[74] S. Nojiri and S. D. Odintsov, “Dark energy, inflation and dark matter from modified F(R)gravity,” arXiv:0807.0685 [hep-th].arXiv:0807.0685 [hep-th].