Wave zone in the Hořava-Lifshitz theory at the kinetic-conformal point in the low energy regime
aa r X i v : . [ h e p - t h ] F e b Wave zone in the Hoˇrava-Lifshitz theory at thekinetic-conformal point in the low energy regime
Jarvin Mestra-P´aez , Joselen Pe˜na , Alvaro Restuccia , Departamento de F´ısica, Universidad de Antofagasta, Aptdo 02800, Chile.
Abstract
We show that in the Hoˇrava-Lifshitz theory at the kinetic-conformal point,in the low energy regime, a wave zone for asymptotically flat fields can beconsistently defined. In it, the physical degrees of freedom, the transversetraceless tensorial modes, satisfy a linear wave equation. The Newtoniancontributions, among which there are terms which manifestly break the rel-ativistic invariance, are non-trivial but do not obstruct the free propagation(radiation) of the physical degrees of freedom. For an appropriate value ofthe couplings of the theory, the wave equation becomes the relativistic onein agreement with the propagation of the gravitational radiation in the wavezone of General Relativity.
Keywords:
Gravitational-Waves, Hoˇrava-Lifshitz, Wave-Zone.
1. Introduction
The detection of gravitational waves has opened a new era in the studyof physics [1]. Multimesenger astronomy will be decisive in the study ofastrophysical and cosmological events and can lead to the discovery of newphenomena in extreme situations beyond the reach of experimental tests thatwe now carry out.Relativistic invariance has been proved experimentally in low energies, inparticle accelerators up at ∼
10 Tev level, and in astro-particles detection upto ∼ Tev, see [2]. It is well known that Standard Model of particles andGeneral Relativity (GR) are incompatible at the Planck scale ∼ Tev. E-mail: [email protected]; [email protected]; [email protected] submitted to Physics Letter B February 9, 2021 o have predictions at that scale it is necessary to have a consistent theoryof quantum gravity, still under development. Although the Standard Modeland General Relativity are formulated with great success on the basis of therelativistic principle, its validity at very high energies should be confirmed.Recent astro-particles experiments suggest that in Actives Galactic Nucleus(AGNs) Lorentz Invariance Violation (LIV) [2] should be tested.Hoˇrava-Lifshitz gravity is a recent proposal for a candidate to ultravioletcompletion of General Relativity [3]. This theory models the gravity as 4-dimensional differentiable manifold with a foliation-structure of co-dimensionone. The foliation-leaves are 3-dimensional Riemannian submanifolds. Inaddition, the time and space scale in different ways, consequently the rel-ativistic symmetry is manifestly broken. The anisotropic scaling allows toinclude interaction terms with high spatial derivatives in the potential, with-out breaking the symmetry of the action under diffeomorphisms that preservethe foliation while keeping the second-order time derivatives of the kineticterm [4]. The theory ends up being power counting renormalizable and uni-tary [5]. The theory contains several coupling constants. There is only onein the kinetic term of the Hoˇrava-Lifshitz action, it is dimensionless andplays a relevant role in the theory. When its value is λ = 1 /
3, the so-calledkinetic-conformal point, the theory, propagates the same degrees of freedomof General Relativity and with an appropriate choice of coupling parameters,it is consistent with low energy experiments [6, 7].Arnowitt, Deser and Misner (ADM) proved in the 60’s that General Rel-ativity (GR) has a well defined wave zone. In this space-time region themetric components g T Tij at order O (1 /r ) satisfy the same wave equation asin perturbation theory around a Minkowski space-time. In the wave zonethere also exists a Newtonian background at order O (1 /r ) that does not pre-vent the g T Tij modes, the physical degrees of freedom, to propagate as freeradiation [8].We prove in this work that in the ( λ = 1 /
2. Foliation, geometry and Hoˇrava-Lifshitz gravity
Let M a 4-dimensional differentiable manifold. M has a codimension-onefoliation structure ( M, F ) if the maximal atlas F ≡ ( U A , ϕ A ) i.e M = ∪ U A ,where U A is a family of open subsets of M and ϕ A : U A → D A ⊂ R × R are diffeomorphism such that if U i ∩ U j = ∅ the transition of charts is definedby ϕ i ◦ ϕ − j : ϕ j ( U i ∩ U j ) → ϕ i ( U i ∩ U j ) , (1)( t, x ) → (˜ t ( t ) , ˜ x ( t, x )) . (2)The couple ( M, F ) and its equivalents under the diffeomorphisms, thatpreserve the foliation structure, F Diff , provide the geometrical structure ofthe Hoˇrava-Lifshitz theory where space and time scale anisotropically t → b z t and x → bx . We remark that M is the disjoint union of 3-dimensional Rie-mannian manifolds (Σ t , g ij ), g ij ( t, x ) = ∂ ˜ x l ∂x i ∂ ˜ x m ∂x j ˜ g lm (˜ t, ˜ x ), where the followinggeometric objects compatibles with the foliation structure are introduced: aproper time defined through the introduction of the lapse N and a shift ofthe spatial coordinates defined through N i in order to have a contravarianttransformation law under F Diff . ˜ N (˜ t, ˜ x ) d ˜ t = N ( x, t ) dt, (3) d ˜ x i + ˜ N i (˜ t, ˜ x ) d ˜ t = ∂ ˜ x i ∂x j [ dx j + N j ( t, x ) dt ] , (4)we emphasize that there is not a space-time metric on M . The metric on theleaves of the foliation g ij and the fields N and N i are used to describe theevolution of the gravitational field. They scale anisotropically as g ij → b g ij , N → b N and N i → b z − N i .Taking into account the anisotropic scaling and foliation structure theproposal incorporates terms with high spatial derivatives in the potentialwithout breaking the symmetry under F Diff . The Hamiltonian of the Hoˇrava-Lifshitz gravity theory at the kinetic-conformal point is given by [9, 10, 11, 7] H = Z Σ t d x (cid:26) N √ g (cid:20) π ij π ij g − V ( g ij , N ) (cid:21) − N j H j − σP N − µπ (cid:27) + βE ADM , (5)3ere π ij is the canonical conjugate of g ij , N i ≡ g ij N j , σ and µ are Lagrangemultipliers. The surface integral (6) E ADM ≡ I ∂ Σ t ( ∂ j g ij − ∂ i g jj ) dS i (6)Φ N ≡ I ∂ Σ t ∂ i N dS i , (7)is added in order to ensure the differentiability of the Hamiltonian, see [12]where this idea was introduced. E ADM is the well known ADM-Energy. Bothsurface terms (6) and (7) appear as surface terms in the Hamiltonian whenit is rewritten in terms of a linear combination of constraints.The potential, up to terms quadratic on the Riemann tensor and thevector field a i , is V = V (1) + V (2) + V (3) , with [5] V (1) = βR + αa i a i , (8) V (2) = α R ∇ i a i + α ∇ i a j ∇ i a j + β R ij R ij + β R , (9) V (3) = α ∇ R ∇ i a i + α ∇ a i ∇ a i + β ∇ i R jk ∇ i R jk + β ∇ i R ∇ i R , (10)where a i ≡ N ∇ i N , ∇ i represents the affine connection, the covariant deriva-tive constructed with the Riemannian metric on the leaves, α ’s and β ’s arecoupling constants. The potential also contains terms of the same order onspatial derivatives as the ones explicitly shown, but of cubic order or greateron the Riemann tensor and the vector field a i , since they do not contributeto the dominant order in the wave zone they are not presented in (9) and(10). The primary constraints of the theory are H j = 2 ∇ i π ij = 0 , P N = 0 , π = g ij π ij = 0 , (11)where only H j = 0 ends up being a first class constraint.If we consider only the low energy potential, V (1) , the time preservationof primary constrains imply the following secondary constraints:
32 1 √ g π ij π ij + √ gβR + √ g (cid:0) α − β (cid:1) a i a i − β √ g ∇ i a i = 0 , (12) √ g ( π ij π ij − βgR ) + α √ ga i a i + 2 α √ g ∇ i a i = 0 , (13)which together with the other two constraints in (11) are second class con-straints. 4he evolution equations are,˙ g ij = 2 N √ g π ij + 2 ∇ ( i N j ) − µg ij , (14)˙ π ij = N g ij √ g π kl π kl − N √ g π ik π j k + N √ gβ (cid:18) R g ij − R ij (cid:19) + β √ g (cid:2) ∇ ( i ∇ j ) N − g ij ∇ N (cid:3) − αN √ g (cid:18) a i a j − g ij a k a k (cid:19) −∇ k (cid:2) π k ( i N j ) − π ij N k (cid:3) + µπ ij . (15)
3. The gravitational fields in the wave zone
We define the wave zone, as in [8], by the following three conditions:First, kr ≫ k is the wave number and r is radial distance. This con-dition is the same that defines the far-zone in linear theories such as classicalelectrodynamics, and can be satisfied if the radial distance is far enough fromthe sources. In addition to the previous condition, for non-linear theories itis necessary to impose more restrictive conditions in order to ensure that theself-interaction does not destroy the free propagation of dynamical modes.Then as a second condition, we demand that the deviations of the fields from”flat background” are of the order of O (1 /r ), i.e. | g ij − δ ij | ∼ | N − | ∼| N i | = O ( A/r ) ≪
1, where A ( t, θ, φ ) represent generic functions of time,and angular coordinates such that A and all its derivatives are bounded.The third condition is | ∂g/∂ ( kr ) | ∼ | ∂N/∂ ( kr ) | ∼ | ∂N i /∂ ( kr ) | ≪ | g − δ | ,which is necessary to guarantee that the interaction of sub-leading ordermodes can not interfere to leading order O (1 /r ).In order to identify which field components propagate and which are staticones, at the leading order, we make an orthogonal linear decomposition intransverse and longitudinal parts [8]. A symmetric tensor that vanishes atinfinity can be expressed as f ij = f T Tij + f Tij + 2 ∂ ( j f i ) . (16)The transverse part f Tij ≡ [ δ ij f T − ∂ i ∂ j f T ] is divergence-free. The f T Tij -partis divergence-free and trace-free. The remaining term 2 ∂ ( j f i ) is its longitu-dinal part. f T = δ ij f Tij is the trace of the transverse part of f ij . is theinverse of the flat space Laplacian, defined on the space of functions whichvanishes at infinity. 5he solution of primary constraints Eq.(11) implies that the propagatingparts of the momenta in the wave zone behaves as π ijT ∼ π ijL ∼ B ij r + k ˆ A ij e ikr r , (17)we use A y B as generic tensorial functions of angles such that they and theirderivatives are bounded. The only propagating part of momentum is π ijT T ∼ B ij r + k ˆ A ij e ikr r . (18)If we use Eq.(17, 18) in the secondary constrains Eq.(12, 13) and thetransverse gauge g ij,j = 0, we obtain a coupled system of second-order el-liptical partial differential equations for variables g T and N . If we multiplyEq.(12) by two and add the result to Eq.(13) we decoupled the system andwe can estimate N and g T in the wave zone: N − ∼ g T ∼ Br + ˆ Ae ikr r . (19)Note that these fields have a Newtonian part O (1 /r ).In the low energy regime, where only the contribution (8) to the potentialis considered, the term αa i a i breaks manifestly the relativistic symmetry. Itscontribution to the present analysis can be determined directly from (19).We obtain a i . B i r + k ˆ A i e ikr r , a i a i . Br + k ˆ Ae ikr r , ∇ i a i . Br + k ˆ Ae ikr r , (20)although terms including the vector a i are involved in the field equationsthey do not contribute to the dominant order O (1 /r ).From the transverse and longitudinal decomposition of the dynamicalequations Eq.(14,15), we obtain the canonical form of the wave equation˙ g T Tij = 2 π T Tij + O (1 /r ) , (21)˙ π ijT T = 12 β ∆ g T Tij + O (1 /r ) , (22)equivalently, ¨ g ijT T − β ∆ g ijT T = 0 + O (1 /r ) . (23)6hen to the leading order O (1 /r ) the transverse traceless components ofthe spatial metric satisfy a wave equation with speed of propagation √ β .Detection from gravitational waves arising from the merge of the neutron starbinary system GW170817 [13] and its electromagnetic counterpart, γ -rayburst GBR170817A [14], restrict the space of β -parameter to | − √ β | ≤ − [15]. Then, if that is so, the prediction on gravitational waves ofHoˇrava-Lifshitz theory at the kinetic-conformal point is the same as in GR.
4. Discussion and conclusions
We showed that in Hoˇrava-Lifshitz theory at the kinetic-conformal point,in the low energy regime, a wave zone can be consistently defined. The phys-ical degrees of freedom, the transverse traceless tensorial modes, satisfy alinear wave equation. The same one that arises from a linear perturbativeapproach [16, 17], but unlike this there are Newtonian non-trivial contribu-tions, that do not obstruct the free propagation (radiation) of the physi-cal degrees of freedom. Among these contributions there are terms whichmanifestly break the relativistic symmetry. These terms which determine adifferent physical behavior of the static, spherically symmetric solutions ofthe Hoˇrava-Lifshitz gravity theory compared to GR, do not contribute to thedominant order in the wave zone.In Hoˇrava-Lifshitz theory there does not exist a universal (scale invari-ant) constant as the light velocity, however the energy-dependent couplingconstant √ β in the renormalization flow from the UV regime to the IR pointshould end up having a value very near or equal to the speed of light. Inthat case, although the Hoˇrava-Lifshitz theory breaks the relativistic sym-metry, the wave equation coincides with the relativistic one arising in GR.We expect that the interaction terms with high spatial derivatives modify thewave equation by introducing linear high order spatial derivatives in the wavezone, i.e ∆ and ∆ operators in the propagating equation. However, it isunknown if in this case the Newtonian background interacts in a non-trivialway with the propagation of the physical degrees of freedom. In particular,the resolution of the constraints is in this case a non-trivial problem.
5. Aknowledgments
J. Mestra-P´aez acknowledge financial support from Beca Doctorado Na-cional 2019 CONICYT, Chile. N ° BECA: 21191442. J. M. P is supported7y the projects ANT1956 and SEM18-02 of the University of Antofagasta,Chile.
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