Stability of warped AdS3 vacua of topologically massive gravity
aa r X i v : . [ h e p - t h ] M a y Preprint typeset in JHEP style - HYPER VERSION
Stability of warped
AdS vacua oftopologically massive gravity Dionysios Anninos , Mboyo Esole and Monica Guica Jefferson Physical Laboratory, Harvard University,17 Oxford St., Cambridge, MA 02138, USA Laboratoire de Physique Th´eorique et Hautes Energies (LPTHE)Universit´e Pierre et Marie Curie-Paris 6; CNRS UMR 7589Tour 24-25, 5 ` eme ´etage, Boite 126, 4 Place Jussieu75252 Paris Cedex 05, France Abstract:
AdS vacua of topologically massive gravity (TMG) have been shown to be perturbativelyunstable for all values of the coupling constant except the chiral point µℓ = 1. We studythe possibility that the warped vacua of TMG, which exist for all values of µ , are stableunder linearized perturbations. In this paper, we show that spacelike warped AdS vacuawith Comp`ere-Detournay boundary conditions are indeed stable in the range µℓ > AdS vacuum. The situation somewhat resembles chiral gravity: althoughnegative energy modes do exist, they are all excluded by the boundary conditions, and theperturbative spectrum solely consists of boundary (pure large gauge) gravitons. Keywords: black holes, topologically massive gravity, warped AdS. ontents
1. Introduction 22. Preliminaries I: the background solution 3
AdS backgrounds 42.3 Warped AdS black holes 52.4 Asymptotic behavior 6
3. Preliminaries II: first order perturbation theory 7
AdS
4. The massive gravitons 10 k = 0 15
5. Boundary gravitons 16
AdS
6. Summary and open questions 17A. Properties of global warped
AdS A.1 Global vs. fibered coordinates in
AdS B. Gauge-fixing 22C. Various expressions 24D. Analysis of the linearized equation of motion 25E. Consistency checks of the boundary conditions 26 – 1 – . Introduction
Topologically massive gravity (TMG) [1, 2] is an interesting extension of three-dimensionalgravity which contains both propagating degrees of freedom as well as black hole solutions.The action of TMG is obtained by adding to the usual Einstein-Hilbert action with a pos-itive Newton constant a gravitational Chern-Simons contribution, with coupling constant1 /µ . As for the usual Einstein-Hilbert action, TMG may also be supplemented with anegative cosmological constant − /ℓ .A particular vacuum of TMG with a negative cosmological constant is AdS , whichalso contains the BTZ black holes [3, 4]. For arbitrary Chern-Simons coefficient, the AdS vacuum suffers from perturbative instabilities. However, it was noted in [5] that at thespecial point µℓ = 1, the AdS vacuum is stable and the theory has a purely chiral spectrum[6]. It has been shown in [7] that the quantum partition function of chiral gravity has allthe required features for the theory to be dual to an extremal CFT [8]. AdS is only one of various possible vacua of TMG with a cosmological constant. Asshown in [9–13], less symmetric vacua known as warped AdS occur as classical solutions tothe equations of motion. These solutions are specific to TMG, i.e. they are not solutionsof pure Einstein gravity with a cosmological constant. Their defining property is thatthey are real line fibrations over AdS preserving a single SL (2 , R ) isometry of the original SL (2 , R ) L × SL (2 , R ) R AdS isometries, along with a non-compact U (1) isometry generatedby translations along the fibre coordinate.The warped vacua of TMG fall into three types: spacelike, timelike and null warped,depending on whether the norm of the Killing vector generating the U (1) isometry ispositive, negative or zero. Each of the first two types can be further classified as stretched( µℓ >
3) or squashed ( µℓ <
3) depending on the magnitude of the warp factor. The factthat these background spacetimes are not asymptotically
AdS makes them very interestingto study, since we could hope to develop new types of holographic correspondences. In [14]such a correspondence was proposed and the central charges of the putative CFT wereconjectured .Quotients of warped AdS along various Killing directions may give rise to black holes[14], in perfect analogy with the BTZ case in AdS . Black hole solutions free of closedtimelike curves (CTCs) can only be found in spacelike stretched and null warped AdS .One can also consider quotients of spacelike warped AdS along the U (1). Such geometrieshave Killing horizons and no CTCs, they resemble the self-dual solutions in AdS [23].Thus, the spacelike warped TMG vacua seem the most interesting to study.The subject of the present article is the classical stability of these spacetimes. Specialattention will be given to the spacelike stretched case, which is richer and better understood.The stability about a certain background depends on the selection of consistent boundaryconditions. For example, the propagating mode of TMG in an AdS background carriesnegative energy, thus rendering the theory unstable. Chiral gravity is spared because the Related questions have been tackled in the case of the Kerr/CFT correspondence [15], and in the recentsubject of theories dual to non-relativistic CFTs [16,17]. Warped
AdS has also been studied in the contextof string theory [18–22]. – 2 –assive graviton disappears from the spectrum at µℓ = 1, for Brown-Henneaux boundaryconditions [24, 25]. There exist several consistent choices of boundary conditions for TMGin AdS [26, 27], but only the more restrictive ones exclude the negative energy modes.We find a similar situation for propagating modes in spacelike stretched AdS . Whilethe massive gravitons of warped AdS have negative energy, we will see that they do notobey the Comp`ere-Detournay boundary conditions [28,29], which are the only consistent setof boundary conditions proposed in the literature. Thus we discard the massive gravitonsfrom the physical spectrum. These boundary conditions are still relaxed enough to allowthe stretched AdS black holes, which was the original reason they were studied.Having discarded the propagating modes from the spectrum, all we are left with arepure (large) gauge modes. It is well known that if these excitations fall off slowly enoughnear the boundary of the spacetime they should be included in the physical phase space.The analysis of the asymptotic symmetry group in spacelike stretched AdS performedin [28] shows that the energy of such pure large gauge modes has to be positive. Incidentally,for our proposed boundary conditions the only remaining excitations have definite chirality.This is quite reminiscent of what happens in chiral gravity.While we find strong evidence for the stability of stretched AdS , the case of squashed AdS remains inconclusive. The main culprit is our lack of understanding of the boundaryof this space and of a set of consistent boundary conditions. Nevertheless, all the explicitand implicit propagating solutions to the linearized TMG equations of motion that wefound hold equally well for squashed as they do for stretched AdS , and they can be usedto study the stability of this spacetime once the aforementioned issue is overcome.The organization of this paper is as follows: in section 2 we review the TMG action,warped AdS backgrounds and black holes. In section 3 we describe our general procedurefor gauge-fixing, finding the linearized solutions to the TMG equations of motion, impos-ing boundary conditions and computing the energy density of the gravitational waves. Insection 4 we display the explicit highest weight propagating solutions and compute theirenergy. We also study general propagating solutions. In section 5 we display the bound-ary conditions for stretched AdS . We conclude with a discussion in section 6. Variousderivations and expressions are presented in the appendices.
2. Preliminaries I: the background solution
In this section we discuss the basic framework and the background warped
AdS geometrythat we will work with. We also review the black hole solutions obtained from discreteglobal identifications of the background and the asymptotic structure of these solutions. The action for topologically massive gravity (TMG) [1, 2] is I T MG = 116 πG Z d x √− g (cid:20) R + 2 ℓ − µ ε λµν Γ ρλσ ( ∂ µ Γ σρν + 23 Γ σµτ Γ τνρ ) (cid:21) (2.1)where ε = +1 / √− g and µ is a dimensionful coupling with dimensions of mass. Whileit is well-known that pure three-dimensional gravity possesses no propagating degrees of– 3 –reedom, the addition of the higher derivative term introduces a new, massive, propagatingdegree of freedom, the so-called massive graviton. When linearizing TMG about flat space,the mass squared of the propagating graviton is µ .It will prove to be convenient to introduce the new quantities ℓ and ν , defined asΛ = − ℓ , µ = 3 νℓ (2.2)In terms of these, the TMG equations of motion are given by R µν − R g µν − ℓ g µν + ℓ ν C µν = 0 (2.3)where C µν = ε µαβ ∇ α ( R βν − g βν R ) (2.4)The Cotton tensor, C µν , is symmetric, traceless and conserved and vanishes on shell for allEinstein solutions. AdS backgrounds All solutions to pure 3d Einstein gravity with a cosmological constant have vanishingCotton tensor, so they are automatically solutions of TMG. Nevertheless, there also existnontrivial solutions particular to TMG, such as the warped
AdS solutions, which come inseveral types [12–14, 30].We will focus on the vacuum solution known as spacelike warped anti-de Sitter space,with metric: ds = ℓ ν + 3 (cid:20) − (1 + r ) dτ + dr r + 4 ν ν + 3 ( dx + rdτ ) (cid:21) (2.5)where r, τ, x ∈ ( −∞ , ∞ ). The boundary of this space resides at r → ±∞ for fixed x and x → ±∞ for fixed r . To simplify our formulae, we will often use the warp factor a ≡ ν √ ν + 3 , a ∈ [0 ,
2) (2.6)If a > a < a = ν = 1 corresponds to global AdS .The above coordinates are geodesically complete, as reviewed in Appendix A. Fur-thermore, the constant τ slices are spacelike for all r , thus rendering τ as our global timecoordinate. When ν ≤ ∂ τ is a globally defined timelike Killing vector; however, for ν > r . Thus for ν > ν > AdS is to notice that for ν = 1, the metric (2.5) describes AdS when written as a Hopf fibration over AdS ds = ℓ (cid:18) − (1 + r ) dτ + dr r (cid:19)| {z } AdS + ℓ dx + rdτ ) | {z } fibre (2.7)– 4 –he six isometries of AdS form the group SL (2 , R ) L × SL (2 , R ) R . The SL (2 , R ) R , whichis generated by the ˜ L ± , ˜ L Killing vectors below, leaves the expressions in each of theparentheses invariant. It is then apparent that upon turning on a = 1, the isometry groupof the space will be SL (2 , R ) R × U (1) L , where U (1) L - which is noncompact - is generatedby x -translations. The Killing vectors are˜ L = i ∂ τ , J = − i ∂ x (2.8)˜ L ± = ± e ± iτ (cid:18) r √ r ∂ τ ∓ i p r ∂ r + 1 √ r ∂ x (cid:19) (2.9)and they obey the usual SL (2 , R ) algebra under Lie brackets:[ ˜ L , ˜ L − ] = 2 ˜ L , [ ˜ L ± , ˜ L ] = ± ˜ L ± , [ J , ˜ L j ] = 0 , j ∈ { , ± } . (2.10)In terms of the original AdS Killing vectors, the SL (2 , R ) R generators are preservedby the warping, together with the J Killing vector of the SL (2 , R ) L isometries. The restare explicitly broken. Note that in spacelike warped AdS there is no Killing vector withcompact orbits: the isometry generated by L − ˜ L , which was such a Killing vector in AdS , is explicitly broken by the warping. AdS black holes As mentioned in the introduction, TMG contains black holes which are locally spacelikestretched
AdS . The metric is given by ds ℓ = dt + d ˜ r ( ν + 3)(˜ r − r + )(˜ r − r − ) + (cid:16) ν ˜ r − p r + r − ( ν + 3) (cid:17) dtdθ + ˜ r (cid:16) ν − r + ( ν + 3)( r + + r − ) − ν p r + r − ( ν + 3) (cid:17) dθ (2.11)where θ is identified by 2 π . The inner and outer horizons are given by r − and r + whichare positive. Note that the above black holes are free of CTCs only for ν ≥
1. Asdiscussed in [14], these black holes are obtained from discrete global identifications ofspacelike stretched
AdS . They are analogues for the case of warped AdS of the BTZblack hole in AdS . In fact, when ν = 1 the metric becomes the BTZ metric in a rotatingframe. We note that slices of constant t and θ are both spacelike for ˜ r > r + . However, themetric is everywhere Lorentzian.The mass and angular momentum are related to r + and r − . There is a continuousspectrum of black holes all the way to r + = r − = 0, where we find warped AdS in Poincar´ecoordinates with τ identified. Following the analogy with BTZ in AdS , we would expectthat lowering the energy below the black hole continuum we should find a mass gap, withglobal stretched AdS being the ground state. This is in fact not the case, and it can beshown that there are no values of r ± for which the metric is both real and has a global SL (2 , R ) × U (1) isometry. Thus, the vacuum (unquotiented) space is not part of the familyof spacetimes (2.11). This is in agreement with the fact that global warped AdS has– 5 –o Killing vectors with compact orbits, a property that the metrics (2.11) clearly do notshare .It has been shown [30] that the black holes (2.11) obey the first law of thermodynam-ics, once one employs the Chern-Simons corrected entropy formula [31–33]. In [14] thisentropy formula has been suggestively rewritten as the entropy of a thermal state in atwo-dimensional CFT with unequal left-and right-moving central charges c R = (5 ν + 3) ℓν ( ν + 3) G , c L = 4 νℓ ( ν + 3) G (2.12) In order to answer questions about boundary conditions, we need to understand where theboundary circle lies in the geometries of interest, since the conserved charges are constructedas integrals over this circle.Let us start with the easy case, which are the warped black holes of the previoussection. The boundary circle consists of two disconnected pieces, one at ˜ r → ∞ and one at˜ r → −∞ , and each is parameterized by θ . Asymptotically, the warped black hole metriccan be written as ds ℓ = 3( ν − r dθ d ˜ r ( ν + 3)˜ r + dt + 2 ν ˜ rdtdθ + h µν dx µ dx ν (2.13)where the ‘perturbation’ h µν falls off at least one power of ˜ r faster than the background [28].The boundary conditions defined near this boundary are reasonably well understood.Note that the asymptotic form of (2.11) and (2.5) are the same, given that one iden-tifies τ ↔ ( ν + 3)2 θ , r ↔ ˜ r , x ↔ ( ν + 3)2 ν ( t − νr θ ) (2.14)Nevertheless, for global warped AdS both τ and x are noncompact, so this identificationonly holds locally at asymptotic infinity. Moreover, it is not quite clear where the boundarycircle lies in these coordinates. If we consider a surface of constant τ - which is alwaysspacelike and always intersects the boundary, we find that the induced metric is ds τ = a dx + dσ , σ = sinh − r (2.15)Thus, the boundary circle consists of four pieces: r → ±∞ , x finite , x → ±∞ , r finite (2.16)This is the same conclusion that one reaches when studying the boundary of AdS in fiberedcoordinates (see appendix A). Quotienting along various Killing vectors [14] gives the blackholes (2.11). These quotients act on the boundary and split it into two disconnected circlesat r → ±∞ . We are grateful to A. Strominger for making this point. We define r = √ r + r − ( ν +3)2 ν . – 6 – . Preliminaries II: first order perturbation theory Having discussed the background geometry and its asymptotic structure, we now delveinto the linearized equations of motion. We also define the notion of energy, for bothpropagating and pure large gauge solutions, that we will use to test stability.
AdS Our goal is to study linearized perturbations around the background (2.5), and find outwhether they can destabilize the spacetime. The linearized equations of motion for a metricperturbation h µν read R (1) µν − ( 12 R + 1 ℓ ) h µν − ℓ ν C (1) µν = 0 (3.1)where R (1) µν = 12 ( ∇ λ ∇ µ h λν + ∇ λ ∇ ν h λµ − ∇ λ ∇ λ h µν − ∇ µ ∇ ν h µµ ) C (1) µν = ǫ µαβ ∇ α ( R (1) βν − h βν R ) − ǫ µαβ δ Γ λαν ( R βλ − g βλ R ) (3.2) δ Γ λµν = ( ∇ µ h λν + ∇ ν h λµ − ∇ λ h µν ) (3.3)All derivatives are taken with respect to the background metric (2.5).When studying perturbations around AdS , the fact that the background is maximallysymmetric drastically simplifies the third order linearized equations of motion. It can thenbe shown [5] that if one chooses to work in harmonic gauge ∇ µ h µν = 0 (3.4)the equations of motion (3.1) take the form D M D L D R h µν = 0, where D I are three com-muting linear differential operators. Of the three distinct solutions that one gets for generic µ away from the chiral point, only one describes the propagating massive graviton, whilethe other two are pure (large) gauge [5]. It is apparent that the condition (3.4) did notcompletely fix the gauge redundancy in the problem.Due to the fewer symmetries of the warped AdS background, we were unable to bringthe equations of motion to an analogously simple form. Nevertheless, we may expect thethird order equations of motion to split at least into a first order piece and a second orderone - the reason being that TMG does describe one propagating mode, which should obeya second order wave-like equation. The most elegant option not readily feasible, we havedecided to attack these equations on two fronts: • fix the gauge completely and solve the linearized equations of motion. In this way,we are sure to only be describing the propagating mode.In agreement with our expectations, the equations of motion decouple . While the equationwe obtain is in principle tractable, for the purposes of this article it is only the asymptoticbehavior of the solution that is relevant, so we leave its full analysis for subsequent work. We have assumed though that we can concentrate solely on separable momentum eigenstates. – 7 – use the SL (2 , R ) R × U (1) L isometry of the background to classify perturbations. Asis well known, linearized solutions to the equations of motion must fall into represen-tations of the isometry group of the background. The most commonly encounteredrepresentation of the above isometry group is the highest weight one. We thus con-sider a basis of perturbations ψ µν that are eigenfunctions of U (1) L and belong to ahighest weight representation of SL (2 , R ) R J ψ µν = kψ µν , ˜ L ψ µν = ωψ µν , ˜ L ψ µν = 0 (3.5)While it is not true that all solutions to the equations of motion can be written as asuperposition of SL (2 , R ) R highest weight states and their descendants, such perturbationsare a physically relevant subclass of solutions, especially from the point of view of theAdS/CFT correspondence. Even though our analysis is not exhaustive, it still provessufficient for the purposes of this article, as will soon become clear. The solutions ofphysical interest can be split into two types: • propagating , if the metric perturbation cannot be written as ψ µν = L ξ ¯ g µν for anydiffeomorphism ξ µ . Here ¯ g µν is the background metric (2.5). • pure large gauge , if ψ µν = L ξ ¯ g µν for some diffeomorphism ξ µ that does not vanish‘sufficiently rapidly’ at asymptotic infinity.The propagating modes are easy to find: one first chooses a gauge such that all thegauge freedom in choosing the metric components is fixed, and then proceeds to look forsolutions to (3.1) which have a wavelike behavior. In appendix B we describe such a gaugefixing for the case in which the gravitational waves have a nontrivial dependence on thecoordinate x . An appropriate gauge-fixing for the x -independent case has been describedin [34].Since we will concentrate our attention on highest weight solutions, we would like tofind a gauge which preserves the highest weight property. This basically requires that wegauge-fix only using diffeomorphisms that commute with ˜ L and have appropriate weightsunder ˜ L and J .Pure large gauge perturbations automatically satisfy the linearized equations of mo-tion. Whether they are physically relevant is determined entirely by the choice of boundaryconditions, as will be reviewed in the next section. The pure large gauge modes are by def-inition those which generically carry nontrivial conserved charges as measured at infinity,and thus are physically relevant. For example, in pure Einstein gravity in AdS , propa-gating modes do not exist and it is precisely the pure large gauge modes that correspond- upon quantization - to states in the dual CFT. A general procedure for computing theconserved charges, finding a set of consistent boundary conditions and determining whichare the ‘large’ gauge transformations is presented below. The quest for consistent boundary conditions for the metric perturbations in a given back-ground spacetime proceeds in several steps:– 8 –. locate the boundary of the background spacetime2. impose boundary conditions on the metric fluctuations at asymptotic infinity3. find all the diffeomorphisms that preserve the boundary conditions4. show that the boundary conditions are consistent, which requires computing all thecharges associated with the asymptotic symmetry generators and showing that theyare conserved, finite and integrable . If infinities are found, one may need to imposeadditional boundary conditions, or altogether change the ones that were originallyproposed. The asymptotic symmetries are defined to be those allowed diffeomor-phisms which have nonzero conserved charges on a generic allowed background.5. find the Dirac bracket algebra of the asymptotic generators, which form the asymp-totic symmetry group (ASG) [24], together with its eventual central extension.The conserved charges Q ξ associated with the asymptotic symmetry generators ξ µ canbe computed in a variety of ways. A particularly nice formalism has been developed in [35].If ¯ g µν denotes the background metric and h µν a perturbation satisfying the boundaryconditions, then the conserved charges Q ξ [ h, ¯ g ] are constructed as surface integrals overthe spacelike boundary ∂ Σ of the ( n -dimensional) spacetime Q ξ [ h, ¯ g ] = Z ∂ Σ K ( n − ξ [ h, ¯ g ] (3.6)where K ( n − is a particular n − h µν . The explicit expression for K (1) for TMG is given in [28].The above expression for the charges is valid for finite (as opposed to infinitesimal) h µν when a certain property called asymptotic linearity holds, which takes the form Q ξ [ h, ¯ g ] = Q ξ [ h, ¯ g + δg ] , ∀ ξ ∈ ASG (3.7)where δg µν is any perturbation of the background metric consistent with the boundaryconditions. This is because the expression (3.6) was derived for linearized perturbations δh µν around a given background, and one needs to integrate over a path in phase spacein order to find the charges associated with finite departures from the background metric.Thus, the general expression for the charges is Q ξ [ h, ¯ g ] = Z γ Dδg Z ∂ Σ K ( n − ξ [ δg, g ( γ )] (3.8)where γ is a path in phase space which connects ¯ g µν to ¯ g µν + h µν . The above integral onlymakes sense if it is independent of the path γ , which reduces to the requirement of chargeintegrability Q ξ [ δh , g + δh ] − Q ξ [ δh , g + δh ] − Q ξ [ δh + δh , g ] = 0 (3.9)for any background metric g µν allowed by the boundary conditions.– 9 –ntegrability needs to be checked for warped AdS , as the theory is not asymptoticallylinear in this background. Besides integrability, one also needs to check finiteness of thegenerators (3.8) and conservation . In order to address the question of stability, we need to compute the energy of the variousgravitational perturbations [36–39] and find its sign. Using the formalism from the previoussection, the energy in question is just the charge associated to the relevant Killing vector( ∂ τ in this case) of the back-reacted gravity solution.In this section we follow the discussion in [7]. If we let ξ µ ∂ µ = ∂ τ and h µν be someperturbation of the background metric ¯ g µν , then it can be shown that the expression forthe energy is Q ξ [ h, ¯ g ] = 116 πG Z Σ ⋆ ( ξ µ E (2) µν [ h (1) ] dx ν ) (3.10)where E (2) µν are the TMG equations of motion at second order in perturbation theory,evaluated on a solution h (1) µν of the linearized equations of motion. Σ is a spatial slice atconstant τ and ⋆ denotes the Hodge star operation. The quantity − E (2) µν ( h (1) ) is sometimesalso called the energy-momentum pseudo-tensor, because it sources the linearized equationsof motion for the second order metric perturbation. It reads E (2) µν = G (2) µν + ℓ ν C (2) µν (3.11)where G (2) µν , C (2) µν are the Einstein and the Cotton tensor, respectively, evaluated to secondorder in the perturbation h (1) µν . Since we are building plane waves along x , what we willactually be computing is the energy density of a gravitational wave in the x -direction, whichis given by E P = 116 πG Z dr √− g g τµ E (2) µν ξ ν (3.12)and try to establish its sign.
4. The massive gravitons
We find explicit highest weight solutions and the asymptotic structure for all solutions tothe linearized equations of motion. The energy density of the highest weight solutions isfound to be negative. However, we discard the modes by imposing boundary conditions.
We consider the following Ansatz for the metric perturbation ψ µν ( τ, r, x ) = f ( r ) e i ( kx − ωτ ) f ( r ) f ( r ) f ( r ) f ( r ) f ( r ) f ( r ) f ( r ) f ( r ) C (4.1) In fact, conservation of the charges Q ξ should hold by construction; however for relaxed boundaryconditions one must check conservation by hand. – 10 –f we first solve the highest weight condition, we find that the functional form of the solutionis completely fixed, up to six constants C i , i = { , . . . , } , subject to rescaling by an overallfactor. The function f ( r ) takes the form f ( r ) = e k tan − r (1 + r ) − ω (4.2)while the ratios of the metric components take a relatively simple form f ( r ) = C + r C + r C f ( r ) = i r ) (cid:0) C − C + 2 r ( C − C − C ) − r C (cid:1) f ( r ) = C + r C f ( r ) = 1(1 + r ) (2 C − C − C + r ( C − C ) − r C ) f ( r ) = i r ( C − C − r C ) (4.3)Next, we gauge-fix in such a way that the highest weight condition is preserved. It turnsout that as long as k ( k + a ) = 0 (4.4)then one can always set C = f ( r ) = f ( r ) = 0 (actually f = 0 follows from the previoustwo conditions). This amounts to setting three of the constants ( C = C = C ) to zero.Next, one plugs the gauge-fixed highest weight perturbation into a subset of the linearizedequations of motion, which can be written as Av = 0 , A ∈ M × , v = C C C (4.5)The determinant of A isdet A = k ( a + k )( k + ( ω − )( a − k + a ( k − − ω + ω )) (4.6)Thus, if det A = 0, the only solution is C = C = C = 0 and we conclude that all highestweight modes of this form are pure gauge. If det A = 0 we get a non-pure gauge mode,obtained when v is in the kernel of A . The constants C , are then determined in terms of C via the equations of motion.Since we are looking for square integrable solutions, ψ µν , and the range of x is infinite, k must be real. For the classical analysis we are performing, we should also take ω to bereal. Thus we obtain a propagating mode only if ω = ω ± ≡ ± s k (cid:18) a − (cid:19) + 54 − a (4.7)For convenience, let us also define ω , ≡ ± r − a ω, k ∈ R , we can distinguish several cases– 11 – a <
1. All values of k are allowed, while ω ∈ ( −∞ , ω ) ∪ ( ω , ∞ ) (4.9)In this particular squashing parameter range, we have 1 < ω < (1 + √ / − √ / < ω < • < a < √ . In order to have ω ∈ R , | k | must be bounded above as k ≤ − a a − a , while ω ≤ ω ≤ ω (4.10)In this parameter range, we have < ω < < ω < . • a > √ . Then there is no highest weight propagating solution to the linearized TMGequations of motion in this background.In summary, a propagating highest weight mode takes the form ψ µν ( τ, r, x ) = 1(1 + r ) ω e i ( kx − ωτ )+ k tan − r f ( r ) f ( r ) 0 f ( r ) f ( r ) 00 0 0 (4.11)with ω given by (4.7), f i ( r ) by (4.3) with C = C = C = 0 and C , given in terms of C in the appendix. Moreover, k is subject to the restrictions mentioned above. Our next task is to compute the energy of these modes using (3.12). Since the physicalmetric perturbation must be real, we take h µν = α ψ µν + α ∗ ( ψ µν ) ∗ (4.12)Also, given that our wave solutions are ∂ x eigenfunctions and the energy is obtained byintegrating over a whole spatial slice at constant τ , the energy of a single k -mode willdiverge. Thus, it is more appropriate to consider the energy density of the modes per unitlength in the x -coordinate. Finite energy configurations are then obtained by generatinglinear combinations of the k -modes with compact support in x .The expression for the energy density is analytically tractable but extremely lengthyand unilluminating. Its salient feature is that it can be written as a rather simple integral: E P = | αC | Z dr e k tan − r (1 + r ) ω +4 8 X n =0 r n b n ( k, a ) ≡ X n =0 I n ( k, a ) b n ( k, a ) (4.13)where we have defined I n ( k, a ) ≡ Z ∞−∞ dr r n e k tan − r (1 + r ) ω +4 , ω = ω ( k, a ) (4.14)– 12 –ne finds that the above integrals are non-divergent if Re [ ω ] > and n ≤
8. Thus weshould only restrict ourselves to the upper branch ( ω + ) of solutions (4.7), given that thelower branch ( ω − ) always has divergent energy density . For n ≤
8, the integrals I n obeya recursion relation(Ω − n + 1) I n = 2 kI n − + ( n − I n − , Ω = 2 ω + 6 (4.15)so we can rewrite the whole expression for the energy in terms of I ( k, a ). It turns outthat for the entire allowed range of k I ( k, a ) is real and positive, whereas the coefficientmultiplying it is always negative. We thus conclude that E P < , ∀ a ∈ (0 ,
2) & ∀ k allowed. (4.16) At first sight, the fact that the energy of the propagating highest weight modes is alwaysnegative may sound discouraging as far as the stability of warped
AdS is concerned.Nevertheless, an important point is that a theory is not only defined by its Lagrangian,but also by boundary conditions to be imposed on the fields.Consider the expression (4.11) for the highest weight modes. The asymptotic behaviorof the metric perturbation is h ττ ∼ r − ω , h τx ∼ r − ω , h τr ∼ r − ω ,h xx ∼ r − ω , h rr ∼ r − − ω , h rx ∼ r − − ω (4.17)Note that as r → ±∞ the perturbation roughly falls off by a factor of r − ω faster than thecomponents of the background metric. A set of boundary conditions for stretched AdS proposed in [28, 29], and further elaborated in section 5.1, require that perturbations falloff at least by one power of r faster than the background metric. Thus we are instructedto only include those highest weight modes which have ω ( k, a ) ≥ AdS , a > ω ( k, a ) <
1. Thus we can conclude that all thehighest weight, negative energy propagating modes are excluded from the spectrum of TMGin stretched
AdS backgrounds . We will see in the following section that we can in factexclude all propagating modes from the spectrum of TMG if a > AdS renders the theory unstable. The reason is that the a > positive energy in our conventions. It has been quite customary in the context The ‘superradiant’ modes with complex ω ± have Re [ ω ] = , so they carry infinite energy density andthus must be discarded. Note that when ω + − ω − ∈ Z then logarithmic asymptotic behavior is also allowed. This occurs instretched AdS only for a < √ and at isolated values of k in squashed AdS . – 13 –f TMG [1, 2] to reverse the sign of Newton’s constant G , which amounts to reversing thesign of the energy. In asymptotically flat space, the only allowed configurations are negativeenergy massive gravitons, and switching the sign of G seems to render them harmless forthe stability of TMG about flat space. The sign of Newton’s constant is fixed by requiringthat the black holes have positive energy, and no further discussion is possible.In the case of squashed AdS , a < ω ( k, a ) >
1. Thus they are allallowed by the boundary conditions (5.1) so long as we take wave packets with compactsupport in x . Nevertheless, it is unclear whether these boundary conditions make sense inthe case of squashed AdS .Finally, for a = 1 - which is just AdS - the negative energy modes have ω = 1 and arethus allowed by the boundary conditions, both (5.1) and Brown-Henneaux . Nevertheless,we already knew that the propagating mode would have negative energy - since the limiting AdS case must have µℓ = 3, so it is away from the chiral point. We will now decouple the linearized equations for the propagating mode. Again, we assumethat the perturbation of interest is an eigenmode of energy and x -momentum, and thus itcan be written as h µν ( τ, r, x ) = e i ( kx − ωτ ) ˜ h µν ( r ). The k = 0 case has already been foundin [34] and will be discussed in the next subsection. Whenever k = 0, we can safely imposethe gauge h µx = 0 (4.19)as shown in appendix B. We therefore consider the metric Ansatz h µν ( τ, r, x ) = e i ( kx − ωτ ) − (1 + r ) g ( r ) g ( r ) 0 g ( r ) (1 + r ) − g ( r ) 00 0 0 (4.20)The coupled system of equations (3.1) decouples as follows g ′′ ( r ) + A ( r ) g ′ ( r ) + B ( r ) g = 0 (4.21)where A ( r ) = P ( r )(1 + r ) P ( r ) , B ( r ) = P ( r )(1 + r ) P ( r ) (4.22)and P n ( r ) are n th degree polynomials in r , whose coefficients depend on k , a and ω .The expressions for P , , ( r ) are given in appendix C. The existence and regularity of thesolutions to (4.21) can be analyzed using Frobenius’ method. For more details we refer thereader to appendix D. All we need from (4.21) is the asymptotic behavior of the solutions.Following appendix D, we know that as r → ±∞ the solutions behave as g ( r ) = 1 r α ∞ X s =0 a s r s (4.23) At a = 1 the boundary conditions (5.1) are more restrictive than Brown-Henneaux boundary conditions,as they only allow half of the usual AdS asymptotic symmetry group. – 14 –here α is a solution to the indicial equation α ( α + 1) − a α + b = 0 (4.24)with a = lim r →∞ rA ( r ) = 2 , b = lim r →∞ r B ( r ) = ( a − k + a a (4.25)It is easy to see that the solutions are simply α ± = ω ± defined in (4.7). Consequently, near r → ±∞ , the solution behaves as g ( r ) ∼ r − ω ± (4.26)The remaining metric components have a similar asymptotic behavior g , ( r ) ∼ r − ω ± .Notice that we recover precisely the asymptotic behavior (4.17) of the highest weight modes.There is one difference though, in that highest weight modes were obliged to have ω = ω ± ( k, a ), whereas no such relation is necessary in the case of general propagating modes.In fact, if we consider an arbitrary but decoupled time dependence for the mode, i.e. h µν ( τ, r, x ) = f ( τ ) e ikx ˜ h µν ( r ), the equations of motion imply the same asymptotic fallofffor r as they did for the energy eigenstates. This agrees with the expectation that wecan construct generic solution by superimposing various highest weight modes and theirdescendants, and possibly modes that belong to different SL (2 , R ) representations .In conclusion, we find that the most general propagating solution to the linearizedequations of motion of TMG has the same asymptotic behavior as the highest weightmodes do. The discussion in section 4.3 still applies and, for stretched AdS , we caninvoke boundary conditions which exclude all propagating modes from the spectrum. Theremaining pure gauge modes form the subject of the next section. k = 0Before we move on to the pure gauge modes, let us make a few comments on propagatingsolutions with k = 0. As emphasized in appendix B and elsewhere, our gauge-fixingcondition does not apply in this case.The appropriate gauge-fixing condition and equations of motion were written downin [34], which studied the problem by dimensionally reducing it to propagation in AdS .The authors found that the propagating mode obeys an equation of the form ✷ φ − m φ = 0 , m = − ν − ℓ (4.27)It is not hard to show that the asymptotic falloff of the solution to this equation is thesame as our (4.17), if we set k = 0. Thus, propagating modes with k = 0 are allowedor disallowed by the boundary conditions just as their k = 0 counterparts. This is to beexpected, as different ways of gauge-fixing should not affect the allowed spectrum of the In the analysis of [40] it was noted that ω obeyed a quantization condition, ω = ω + + n, n ∈ Z + . Thisquantization condition stemmed from the requirement that the solution be well-behaved near the originof AdS . Due to the complexity of our equations, we have been unable to obtain a similar quantizationcondition, although it is likely that it exists. – 15 –heory . One further check that the equation (4.27) of [34] and our results indeed agree,is to compare the conformal weight and mass of the k = 0 graviton. Consider a scalarfield of mass m which propagates in spacelike warped AdS . We look for a highest weightsolution of the form Φ( τ, r, x ) = e i ( kx − ωτ ) φ ( r ) (4.28)The conformal weight of the scalar is determined in terms of the mass m as ω = 12 + s k (cid:18) a − (cid:19) + 14 + L m , L = ℓ √ ν + 3 (4.29)Comparing the above result with ω + , we note that the graviton behaves as a scalar fieldof mass m = − ν − ℓ (4.30)This coincides exactly with the result of [34] for the k = 0 case.
5. Boundary gravitons
Here we propose a set of boundary conditions for global stretched
AdS , which are a slightlymodified version of those put forth in [41] for the asymptotic black hole spacetime (2.13). AdS The boundary conditions we use for the asymptotic metric (rescaled by ( ν + 3) l − ) atlarge r are g ττ = ( a − r + r h ττ + O ( r ) , g τr = r − h τr + O ( r − ) g τx = ar + h τx + O ( r − ) , g xx = a + r − h xx + O ( r − ) ,g xr = O ( r − ) , g rr = r − + r − h rr + O ( r − ) (5.1)The perturbations h µν ( τ, x ) of the background metric would generically yield nontrivialconserved charges, while the terms written as O ( r n ) do not contribute to the conservedquantities.These boundary conditions were developed in [28, 29] for the black hole metrics, where τ is identified and the boundary lies at large r only. As noted in section 2.3, globalstretched AdS is not part of the black hole phase space, and the boundary has a piecethat lies at r finite and x → ±∞ . Therefore, the boundary conditions listed above mustbe supplemented by conditions on the falloff of the metric components at large x . We havenot studied precisely what these restrictions look like, but we always consider wave packetswhich die off sufficiently fast as x → ±∞ . In the case of the propagating modes, one can check whether the falloff of the gauge-invariant quantitiesrespects the boundary conditions. For example, a falloff as | x | − at large | x | seems more than sufficient to ensure the finiteness of thecharges. – 16 –ollowing our checklist from section 3.2, we need to make sure that the above boundaryconditions yield charges which are finite, integrable and conserved. A first thing to note isthat TMG with the above boundary conditions is not asymptotically linear. Thus, integra-bility, finiteness and conservation of the charges for all finite h µν that obey the asymptoticequations of motion must all be considered. This has been rigorously done in [41] forasymptotically warped black hole geometries. As we show in appendix E, for fast enoughasymptotic falloff at large | x | the charges do not gain ay additional contributions from theintegrals along x or r , so consistency of the boundary conditions for global stretched AdS follows from consistency of the very related boundary conditions for the black holes. In this section we will be exclusively working in the warped black hole coordinate system(2.11). Given that the boundary conditions (5.1) have excluded all propagating modes fromstretched
AdS , all the remaining physical excitations in our theory must correspond topure large gauge modes: diffeomorphisms that do not fall off fast enough near the boundaryat ˜ r → ∞ . In [28], they were shown to take the form ξ θ = f ( θ ) + O ( 1˜ r ) , ξ ˜ r = − ˜ rf ′ ( θ ) + O (1) , ξ t = g ( θ ) + O ( 1˜ r ) (5.2)As reviewed in section 3.2, to each non-trivial large diffeomorphism there corresponds agenerator of the asymptotic symmetry group. By expanding f ( θ ) , g ( θ ) in Fourier modes,[28] found the ASG to consist of one copy of the Virasoro algebra and a U (1) Kaˇc-Moodyalgebra. The Virasoro acquires a central extension, with positive central charge c R = (5 ν + 3) ℓν ( ν + 3) G >
AdS with the boundary conditions(5.1) is stable .For ν < θ cannot be identified in the coordinates (2.11), as the resulting spacetimewould have CTCs. We can thus no longer use the boundary conditions of [28]. Theboundary conditions (5.1) could in principle still hold, since our coordinate τ is noncompact.Thus, if (5.1) are extendable to a full set of consistent boundary conditions for squashed AdS , then we conclude that the latter spacetime is unstable , since the negative energypropagating modes are not excluded. Whether such an extension is possible for squashed AdS is very unclear.
6. Summary and open questions
In this note, we have addressed the issue of the stability of spacelike warped
AdS . We havefound explicit propagating massive gravitons living in the highest weight representation ofthe SL (2 , R ) R isometry group and the asymptotic falloff of all propagating momentumeigenstates. – 17 –ur highest weight solutions obey an equation of the form (cid:20)
12 ( L + L − + L − L + ) − L + (1 − a ) a ∂ u (cid:21) ψ µν = − (1 − a ) ψ µν (6.1)We have not managed to recover such an elegant equation in terms of the quadratic Casimirfrom the general linearized equations of motion. The only exception is for k = 0, whereit was obtained by [34]. We suspect, however, that under the appropriate gauge choiceand field redefinition this is possible for all values of k . The fact that we have obtaineda decoupled second order wave equation for a single component of the perturbation isevidence in this direction.The explicit solutions we have obtained have negative energy density, however for a > AdS . We take this to be strong evidence that TMGhas a stable set of vacua for a much larger range of the Chern-Simons coefficient, i.e. µℓ > µℓ = 1. To fortify this claim, it would be interesting to explorethe possibility of a positive energy theorem in the context of warped AdS in TMG [42–44].If stability indeed holds, one might suspect that an initial AdS configuration in TMG,which is known to have perturbative instabilities, decays to stretched warped AdS .We should point out that the boundary analysis in [28] only gave rise to a Virasoroextension of the right-moving isometry group, together with a U (1) Kaˇc-Moody algebra.In particular, it did not give rise to the left moving, centrally extended, Virasoro proposedin [14]. It would be very interesting if new consistent boundary conditions were found forwarped AdS , which yield two copies of the Virasoro algebra as the ASG, with the ex-pected central charges. Using the explicit negative-energy solutions found herein, it shouldbe possible to immediately check the stability of warped AdS with the new boundaryconditions.One of the puzzling features of global stretched AdS is that it does not clearly havethe same asymptotic structure as the stretched AdS black holes. It is unclear thereforewhat the meaning of a partition function for stretched AdS is in this case, since one wouldlike to sum over configurations with the same asymptotic behavior. It is possible that onewould need to replace the global vacuum by the r + = r − = 0 black hole.We should also point out that stretched AdS could still be unstable at a nonpertur-bative level.Another open question is whether there exist boundary conditions under which thenegative energy propagating solutions can be discarded in the squashed warped AdS vacua. If we extrapolated the proposal in (5.1) we would conclude that the solutions aretrue instabilities. On the other hand, a different set of boundary conditions resemblingthose discussed in [15] may lead to an exclusion of all the propagating solutions for thesquashed case as well.We have also noted the propagating solutions exhibit some interesting properties intheir own right. For instance there is a qualitative difference between the conformal weightof the stretched and squashed solutions. In particular, for the squashed solutions the– 18 –onformal weight is real for all allowed values of ν and k , whereas in stretched AdS itis only real for a small window of parameter space given by k < (5 − a ) a a − . When theweight becomes complex, the solutions propagate in the r -direction as well and thus thereis a flux of energy escaping the boundary. This is highly reminiscent of the behavior of ascalar field in the near horizon geometry of the extremal Kerr black hole as was studiedin [45]. In fact, [45] noted that this behavior was related to the superradiance of rotatingblack holes and we suspect a qualitatively similar phenomenon might be occurring in thestretched warped AdS background, had we not discarded the solution.It may also be worth noting that the mass squared of the massive gravitons (4.30)becomes negative for the stretched case. Furthermore, when the warp factor satisfies a > /
4, the frequency of the highest weight solutions becomes complex for all values of k . It would thus be interesting to explore the behavior of the theory in these regimes.Finally, we would like to point out that for the squashed case there is another candidatefor a potentially stable vacuum given by timelike squashed AdS . This spacetime is relatedto spacelike squashed AdS by an analytic continuation in the coordinates given by τ → ix and x → − iτ . Thus our expressions for the highest weight solutions are closely relatedto the ones we have obtained. On the other hand, there are no black hole solutions orconsistent boundary conditions known for this spacetime and it would be interesting toexplore such questions. It would also be interesting to study the stability of null warped AdS whose identifications also give rise to black holes. Acknowledgements
We would like to thank F. Denef, O. Dias, G. Giribet, T. Hartman, C. Keeler, H. Reall,W. Song and A. Strominger for illuminating discussions. We especially thank G. Comp`ereand S. Detournay for useful discussions and checks of the boundary conditions. D.A. andM.G. would also like to thank the ESI for its kind hospitality while part of this workwas completed. D.A. and M.E. have been partially funded by a DOE grant DE-FG02-91ER40654.
A. Properties of global warped
AdS We review the geometry of global
AdS expressed as a Hopf fibration over AdS and showgeodesic completeness of spacelike warped AdS . A.1 Global vs. fibered coordinates in
AdS The simplest way to picture
AdS is as a Lorentzian hyperboloid embedded in R , . If thecoordinates on Minkowski space are X , X , X , X , then AdS is the surface X − X − X + X = 1 (A.1)Different coordinate systems are obtained via different embeddings. To obtain AdS in theusual global coordinates ds = − cosh ρ dt + dρ + sinh ρ dφ (A.2) We have set the
AdS radius ℓ = 1. – 19 –ith φ ∼ φ + 2 π , we use the following parametrization X = cosh ρ cos t , X = sinh ρ sin φX = cosh ρ sin t , X = sinh ρ cos φ (A.3)On the other hand, to obtain AdS in the fibered coordinates ds = 14 (cid:2) − cosh σ dτ + dσ + ( dx + sinh σ dτ ) (cid:3) (A.4)we use the following embeddings X = cos τ x σ τ x σ X = sin τ x σ − cos τ x σ X = − cos τ x σ − sin τ x σ X = − sin τ x σ τ x σ − X X − X X ) = sinh σ = sin( t − φ ) sinh 2 ρ (A.6) − X X + X X ) = cosh σ sinh x = − cos( t − φ ) sinh 2 ρ (A.7)In the remainder of this section, we review the structure of the global AdS boundary inthe coordinates (2.5) [23, 46, 47], in the hope that they will help the reader improve his orher intuition about this kind of spaces.In usual global coordinates (A.2), the boundary is at ρ → ∞ and is famously a cylinder,parameterized by t and φ . We would like to find out where the boundary lies in terms ofthe fibered coordinates (A.4) which are also global, but both τ and x are noncompact.Using (A.6) we see that in order to reach the boundary at ρ → ∞ , we must take σ → ±∞ whenever t = φ mod π . Thus the boundary cylinder of AdS is parsed by nullstrips of σ → ∞ and σ → −∞ , as shown in figure 1.Next, we have to understand what happens on the null lines on the boundary t = φ mod π . Note first that all σ = constant hypersurfaces end on these null lines on theboundary. From (A.6) and (A.7) we conclude thatsinh x = − cos( t − φ ) sinh 2 ρ q ( t − φ ) sinh ρ (A.8)Thus, as we increase t (with φ fixed), x varies from −∞ to ∞ on the pink ( σ → ∞ ) strips,while it varies from ∞ to −∞ on the blue ( σ → −∞ ) strips. Note that if we would like to– 20 – igure 1: A chromatic depiction of the way the fibered coordinates cover the
AdS boundarycylinder. Blue strips have σ → −∞ while pink strips have σ → + ∞ . go around the φ circle at fixed t (say 0 < t < π ), we first fix σ → ∞ and take x → −∞ ,then fix x at this value and take σ → −∞ , then fix σ and take x from −∞ to ∞ , then crossback to the σ → ∞ strip while keeping x → ∞ . Thus it is quite a bit more complicated todescribe the compact direction on the AdS boundary in the fibered coordinates.In conclusion, in terms of the fibered coordinates (A.4), the boundary of AdS consistsof the two apparently disconnected pieces at σ → ±∞ , but which are in fact connected at x → ±∞ into the expected boundary circle. A.2 Geodesic completeness
We review the geodesic completeness of warped
AdS [45, 46] . Note that for a = 1, ourcoordinates have been proven in [23] to be complete, i.e. they parameterize in a one-to-onefashion the full embedded hyperbola. It is also clear that the coordinates are global for a = 0.The two conserved quantities associated to the Killing vectors ξ µ ( x ) = ∂ x and ξ µ ( τ ) = ∂ τ are given by dx µ ( λ ) dλ ξ ( x ) µ = g xx dx ( λ ) dλ + g τx dτ ( λ ) dλ = − p (A.9) dx µ ( λ ) dλ ξ ( τ ) µ = g τx dx ( λ ) dλ + g ττ dτ ( λ ) dλ = − e (A.10)where λ is the affine parameter and p and e are constants. The equation ds = 0 determinesthe equation obeyed by r ( λ ): p − e a + 2 epa r ( λ ) − p ( − a ) r ( λ ) + L a r ′ ( a ) = 0 (A.11) Geodesic completeness means that all geodesics are extendible to arbitrarily large positive and negativevalues of their affine parameter. In particular, boundary points are only reached at infinite affine parameter. – 21 –here L ≡ ℓ / ( ν + 3). The null geodesic equation can be solved exactly for r ( λ ). For a = 1 and p = 0 we find in the limit of large rr ( λ ) ∼ e ± √ p − a L a λ . (A.12)Given that r spans the whole real line, it is clear that the points at infinity are not reachedfor any finite value of the affine length. Also interesting is the fact that for a < x -direction are confined within r < ∞ [46].We can also examine the equations obeyed by τ and xτ ( λ ) = 1 B Z λλ dη ( e − pr ( η ))1 + r ( η ) (A.13) x ( λ ) = 1 L Z λλ dη (cid:18) − pL a + r ( η )( − e + pr ( η ))1 + r ( η ) (cid:19) (A.14)Since there are no poles in the integrands, one sees that the infinities of u and τ are notreached for any finite value of λ .Timelike geodesics obey the equation ds = − dλ , which leads to p − e a + L a + 2 epa r ( λ ) + (cid:0) − p ( − a ) + L a (cid:1) r ( λ ) + L a r ′ ( λ ) = 0 (A.15)Once again, the solutions found asymptotically to be of the form r ( λ ) ∼ e ± √ p − a − L a L a λ . (A.16)For a < a > p > L a / ( a − AdS . For the special value of p = L a / ( a −
1) with a > r ( λ ) ∼ λ (A.17)In this case too, however, the boundary is reached only at infinite affine parameter. Thus,our spacetime is timelike and null geodesically complete. B. Gauge-fixing
In this appendix we present a careful derivation of the gauge-fixing condition h µx = 0,which holds for all modes with k = 0.Let us write the background 3d metric as ds = g µν dx µ dx ν + a ( dz + A µ dx µ ) (B.1)where for the rest of the section µ ∈ { , } and ds = − (1 + r ) dτ + dr r , A = rdt (B.2)– 22 –he inverse metric reads g MN = g µν − A µ − A ν a − + A ! (B.3)while the 3 d Christoffel symbols are (3) Γ ρµν = (2) Γ ρµν + a g ρσ ( A ν F µσ + A µ F νσ ) , F µν = ∂ µ A ν − ∂ ν A µ = L ǫ µν (B.4) (3) Γ zµν = − (2) Γ ρµν A ρ − a A σ ( A ν F µσ + A µ F νσ ) + ( ∂ µ A ν + ∂ ν A µ ) (B.5)Γ λzµ = a g λρ F µρ , Γ zzµ = a A λ F λµ , Γ λzz = Γ zzz = 0 (B.6)Next, the strategy is as follows: we consider small perturbations of the background metric g MN = ¯ g MN + h MN (B.7)and expand them in Fourier modes in zh µν ( x, z ) = Z dk h ( k ) µν ( x ) e ikz , h µz ( x, z ) = Z dk a ( k ) µ ( x ) e ikz h zz ( x, z ) = Z dk φ ( k ) ( x ) e ikz (B.8)We would now like to gauge-fix these perturbations. Under a general diffeomorphism, toleading order, the perturbation h MN transforms as δ h MN = ∇ M ξ N + ∇ N ξ M (B.9)The action of the above diffeomorphisms on the Fourier modes of the metric perturbationis δh ( k ) µν = ∇ µ ξ ( k ) ν + ∇ ν ξ ( k ) µ − a ( A ν F µσ + A µ F νσ )( ξ σ ( k ) − A σ ξ ( k ) z ) − ( ∇ µ A ν + ∇ ν A µ ) ξ ( k ) z (B.10) δa ( k ) µ = ∂ µ ξ ( k ) z + ik ξ ( k ) µ − a F µλ ( ξ λ ( k ) − A λ ξ ( k ) z ) , δφ ( k ) = 2 ik ξ ( k ) z (B.11)where ξ ( k ) M are the Fourier modes of the diffeomorphisms. Now we turn to gauge fixing. Itis quite easy to see that we can set φ ( k ) = a ( k ) µ = 0 , k = 0 (B.12)by fixing the corresponding modes of the diffeomorphisms. In this case, h ( k ) µν is gauge-invariant.We need to use a slightly different gauge-fixing for the zero-modes. φ (0) is clearly gaugeinvariant, while a (0) µ can also be set to zero. The residual gauge transformations satisfy ξ λ (0) = A λ ξ (0) z + 1 a L ǫ λµ ∂ µ ξ (0) z (B.13)– 23 –ne can use this residual gauge freedom to set the trace of h (0) µν to zero. Indeed δh = 2 ∇ µ ξ µ (0) − a A µ F µν ξ ν (0) − ∇ µ A µ ξ (0) z = − a A µ F µν ξ ν (0) (B.14)which we can set to zero by choosing ξ (0) µ = ξ (0) z A µ . All gauge freedom is fixed this way.We therefore conclude that a completely gauge-fixed form of the perturbation for k = 0 is h MN ( x µ , k ) = e ikz h ( k ) µν
00 0 ! (B.15)while for k = 0 we could have h MN ( x µ ,
0) = h (0) µν φ (0) ! , tr h (0) = 0 (B.16)or alternatively use the gauge employed in [34]. C. Various expressions
The expressions for the coefficients C , which enter the solution of the highest weightpropagating modes read C = ( − a )( − ω )( − − a + ω ) C − a ( − ω ) + ω − a ( − ω + ω ) (C.1) C = − a √− a ( − a + ω (2 + (3 − ω ) ω ) + a (1 + ω ( − ω ))) C p − a ( − a + ( − ω ) ω ) ( − a ( − ω ) + ω − a ( − ω + ω )) (C.2)The expressions for the polynomials P , , which enter into the linearized equations ofmotion for the propagating modes are P ( r ) = a ω ( a − a k + a ω + k ( − ω )) +2 a kω ( a − a (1 + 4 k ) + 2 a ( − ω ) + 2 k (1 + ω )) r +( a ( − a ) k ( − a + a + k − a (1 + 4 k )) + a ( a − a k + 6 k + a (1 + 6 k )) ω ) r +2 a k ( a ( − − a + a ) + (3 − a + 2 a ) k + 2 k ) ωr + a k ( − k + a (1 + 2 a − a + a + (6 − a + a ) k + k )) r (C.3) P ( r ) = − a kω ( a − a (1 + 4 k ) + 2 a ( − ω ) + 2 k (1 + ω ))) − a (( − a ) k ( − a + a + k − a (1 + 4 k )) + a ( a − a + 3 k + 8 a k + 6 a k + 6 k ) ω − a ( a + k ) ω )) r +2 a kω (2 a − a (1 + 4 k ) + a ( − − k + 10 ω ) + k (1 − k + 10 ω )) r +4 a ( − k (1 + k )( a + k ) + ( a − a k + 6 k + a (1 + 6 k )) ω ) r +6 a k ( a ( − − a + a ) + (3 − a + 2 a ) k + 2 k ) ωr +2 a k ( − k + a (1 + 2 a − a + a + (6 − a + a ) k + k )) r (C.4)– 24 – ( r ) = a ω ( a − k (1 + ω ) + a (3 + 2 ω ) − a k (3 + 4 ω ] ) + a k ( − k + 2 ω + ω ) + a (3 k + 3 ω + ω ))+2 a kω (( − a )( a + k )( a + a + 2 k ) + 2( − a + 2 a + 2 a k − k − a (1 + 4 k )) ω + 3 a ( a + k ) ω ) r +(( − a ) k ( a + 4 a − a + a + a (2 + 3 a ) k + ( − a ) k ) + a ( a ( − a + 2 a ) + 2 a (1 + 6 a ( − a )) k − − a + 12 a ) k − k ) ω + a (2 a (4 + a ) + a (7 − a + 15 a ) k + ( − a ) k ) ω ) r +2 a kω (2 a − k (4 + k + ω ) + a ( − k + 4 ω ) + a (1 − k − ω + k (10 − ω ))+2 a (3( − ω ) + k ( − ω )) + 2 a k ( − ω + k (4 + 5 ω ))) r +( − k ( a + k )( − k + a (2 + 4 a − a + 7 a − a + (10 − a + 4 a ) k + k )) + a ( a (1 + a ) + a (5 + 5 a − a + 12 a ) k +( − a − a + 15 a ) k + 3( − a ) k ) ω ) r + 2 a k ( a ( − a ) (1 + a ) + a ( − a − a + 4 a ) k + ( − a − a + 3 a ) k + ( − a ) k ) ωr +( − a ) k ( a + k )( − k + a (1 + 2 a − a + a + (6 − a + a ) k + k )) r (C.5)The asymptotic form of the ratios of these polynomials that appear in the equations ofmotion is A ( r ) = 2 r + O ( r − ) , B ( r ) = ( a − a + k ) a r + O ( r − ) (C.6) D. Analysis of the linearized equation of motion
In section 4.4 we have obtained the equation of motion for the propagating mode of TMG,which takes the following form d wdz + f ( z ) dwdz + g ( z ) w = 0 (D.1)The variable z is real in our case, but it could in principle be complex. The above differentialequation is said to have a regular singular point at z = z if f ( z ) , g ( z ) are not analytic at z , but ( z − z ) f ( z ) and ( z − z ) g ( z ) are [48].If the differential equation only has regular singular points (or no singular points atall, which occurs when f ( z ) , g ( z ) are analytic in the whole domain of definition), thensolutions to these equation can be constructed. If the singularities of f and g are worsethan above then the equation is said to have irregular singular points and oftentimes thesolutions cannot be found.In our case, as long as P does not have roots of multiplicity more than one which donot occur concomitantly with roots of P , the differential equation has only regular singularpoints. The behavior of the solution in the neighborhood of such a point is given by (weset z = 0 for simplicity) w ( z ) = z α ∞ X s =0 a s z s (D.2)– 25 –here α is called the exponent or index of the singularity and satisfies the equation α ( α −
1) + f α + g = 0 (D.3)where f , g are the constant terms in the Taylor expansion of zf ( z ) , z g ( z ) around z = 0 zf ( z ) = ∞ X s =0 f s z s , z g ( z ) = ∞ X s =0 g s z s (D.4)The expression (D.2) for the solution to the differential equation has radius of convergenceequal to the radius of convergence of the Taylor expansions (D.4). There are two solutions,corresponding to the two roots of the indicial equation (D.3). If the solutions coincide ordiffer by an integer, one of the solutions acquires a logarithmic term.The regular singular point can also occur at infinity, in which case we change variablesto u = z − . The equation (D.1) can be rewritten as d wdu + p ( u ) dwdu + q ( u ) w = 0 (D.5)with p ( u ) = 2 z − z f ( z ) , q ( u ) = z g ( z ) (D.6)The coefficients that now appear in the indicial equation are the constant terms in theexpansion of 2 − zf ( z ) and z g ( z ) as z → ∞ . One can easily show that if α is a solutionof the following indicial equation α ( α + 1) − p α + q = 0 (D.7)where zf ( z ) = ∞ X s =0 p s z s , z g ( z ) = ∞ X q s z s (D.8)then the solution asymptotically takes the form w ( z ) = 1 z α ∞ X s =0 a s z s (D.9)In conclusion, what must be done is to find the zeroes of P ( r ), make sure they do not leadto poles of multiplicity higher than one in A ( r ), and match the solutions on the differentpatches. E. Consistency checks of the boundary conditions
As reviewed in section 3.3, the conserved charges in the Barnich-Brandt formalism [35]are constructed as boundary integrals of a given one-form, K µ which is determined by theequations of motion of the theory. For TMG, the appropriate form of K µ = ǫ µνρ F νρ wasgiven in [28]. In this section, we consider the charges Q ξ = Z ∂ Σ K µ dx µ , (E.1)– 26 –here ∂ Σ is the boundary of global warped
AdS . We reduce the questions of their integra-bility, finiteness and conservation to the question of consistency of the boundary conditionsfor the black hole spacetimes (2.11). The latter is answered in the affirmative in [41]. Sincewe do not know what precise contour the boundary circle of global warped AdS follows,our strategy is to analyze each component of the integrand K µ . While we do restrictourselves to linear perturbations around global warped AdS , we believe that the sim-ple structure we find also holds for perturbations around any background allowed by theboundary conditions (5.1).The asymptotic symmetry group is generated by diffeomorphisms of the form ξ τ = f ( τ ) , ξ r = − rf ′ ( τ ) , ξ x = F ( τ ) (E.2)Let E µν ( τ, x ) be the linearized equations of motion for the asymptotic perturbation h µν ,and let E ( x ) µν ( τ, x ) be its indefinite integral with respect to x : ∂ x E ( x ) µν ( τ, x ) = E µν ( τ, x ). Wefind the following very simple structure K τ = r · f ( τ ) E ( x ) τr + O ( r ) K x = f ( τ ) E rr + ∂ x (something) + O ( r − ) K r = O ( r − ) (E.3)The equations of motion imply that the divergent part of K τ is simply an integrationconstant which is in fact zero [41].If we impose restrictive enough boundary conditions at large x , it is clear that K x and K r will in fact vanish since they only get contributions from | x | → ∞ , which are eliminatedby our unspecified boundary conditions. The expression for the charges is then identical tothe warped black hole case, except for the range of the τ integration. Consistency of ourboundary conditions immediately follows, since the charges have been proven to be finite,integrable and conserved at the full nonlinear level for the warped black hole case.This analysis strongly points towards the consistency of the boundary conditions (5.1)for an appropriate falloff at large | x | . However, one must still prove the integrability ofthe charges and show that similar simplifications occur for the full asymptotic equationsof motion. Also, it is not clear how restrictive the boundary conditions in | x | can be.In particular, it is not a priori clear whether wave packets are allowed by our boundaryconditions for all time: if we start with a wave packet localized in the | x | direction, it mayspread to large x in time . While wave packets in stretched AdS are already excluded bytheir r -falloff, it would be nice to better understand the boundary conditions at large | x | . References [1] S. Deser, R. Jackiw and S. Templeton, “Three-Dimensional Massive Gauge Theories,” Phys.Rev. Lett. , 975 (1982).[2] S. Deser, R. Jackiw and S. Templeton, “Topologically massive gauge theories,” Annals Phys. , 372 (1982) [Erratum-ibid. , 406.1988 APNYA,281,409 (1988APNYA,281,409-449.2000)]. We are grateful to G. Comp`ere for pointing this out. – 27 –
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