Critical points of the Exotic Massive 3D Gravity
CCritical points of the Exotic Massive 3D Gravity
Gaston Giribet , Andr´es Goya , Edmundo Lavia , , Julio Oliva Physics Department, University of Buenos Aires & IFIBA-CONICET,Ciudad Universitaria, pabell´on 1, 1428, Buenos Aires, Argentina Acoustic Propagation Department. Argentinian Navy Research Office (DIIV),Laprida 555, 1638 Vicente L´opez, Buenos Aires, Argentina UNIDEF (National Council of Scientific and Technical Research – Ministry of Defense) and Physics Department, University of Concepci´on, Casilla 160-C, Concepci´on, Chile ∗ Abstract
Exotic Massive 3D Gravity (EMG) is a higher order generalization of Topologically MassiveGravity. As in other theories of this sort, the conserved charges associated to the asymptoticdiffeomorphisms that preserve the boundary conditions in AdS spacetime span two copies of theVirasoro algebra with non-vanishing central charges. Here, we discuss the values of these centralcharges and the corresponding conformal anomaly in relation to the phase space of the theory. ∗ [email protected], [email protected], edmundolavia@fibertel.com.ar, [email protected] a r X i v : . [ h e p - t h ] O c t . INTRODUCTION Chiral Gravity (CG) [1] is a parity-odd theory of massive gravity in 3 dimensions aroundAnti-de Sitter (AdS) space whose mass parameter takes a value such that either the left-moving or the right-moving central charge of the dual conformal field theory (CFT) vanishes.The vanishing of one of the central charges is usually associated to the emergence in thebulk of a massless graviton mode, which produces a long range interaction characterized bya logarithmic fall-off near the boundary [2, 3] h ∼ log( r ) . (1)This log-mode has negative energy in the bulk, and it makes the dual CFT to be non-unitary. This is the reason why, in order to define CG in a consistent way, one needs toimpose strong asymptotically AdS boundary conditions that suffice to eliminate modes like(1) [4]. These boundary conditions are the Brown-Henneaux asymptotic conditions, i.e. thesame as in general relativity (GR) [5]. If such boundary conditions are imposed, then thedual theory turns out to be a chiral CFT .CG was originally formulated as a particular case of Topologically Massive Gravity(TMG) [6, 7] with negative cosmological constant. However, it can be easily generalizedby adding to the TMG field equations other contributions, also representing sensible mas-sive deformations of Einstein equations, such as New Massive Gravity (NMG) [8, 9], MinimalMassive Gravity (MMG) [10], or the recently proposed Exotic Massive Gravity (EMG) [11],all these being particular cases of a more general set of models [12]. In a series of recentpapers [13–16], EMG coupled to TMG around AdS was studied and the properties of itsdual CFT were analyzed (see also [17–20]). In particular, the values of the central chargeswere obtained and the special features the theory exhibits when those charges vanish werestudied. This is the problem we want to revisit here.2 I. EXOTIC MASSIVE GRAVITY
In the metric formalism , EMG is defined by the following field equations [11] R µν − Rg µν + Λ g µν + 1 µ C µν = T µν , (2)where the Cotton tensor is C µν = 12 ε αβµ ∇ α (cid:18) R βν − g βν R (cid:19) + 12 ε αβν ∇ α (cid:18) R βµ − g βµ R (cid:19) , (3)and where T µν = 1 m ε αβµ ∇ α C βν − m ε αβµ ε γσν C αγ C βσ . (4)The limit m → ∞ of this theory leads to TMG, and the limit µ → | µ | < ∞ the theory does not have a definite parity since while GRand the Exotic terms are parity-even, the Cotton tensor is parity-odd.Equations (2) do not follow from a variational principle as they are covariantly conservedonly on-shell; see [12] for details about this mechanism.As in the case of other massive deformations of 3-dimensional Einstein theory, the con-served charges associated to the asymptotic diffeomorphisms in AdS span two copies of theVirasoro algebra [5]; namely[ L ± m , L ± n ] = ( m − n ) L ± m + n + c ± m ( m − δ m + n, , (5)with [ L + m , L − n ] = 0. The central charges c ± , according to the computation of [16], are givenby c ± = 3 (cid:96) G (cid:20) − (cid:96)m µ ± (cid:18) m µ − (cid:96) m (cid:19)(cid:21) , (6)where (cid:96) = 1 / √− Λ is the radius of AdS . This result for c ± differs from the one obtained in[14]. In particular, the charges (6) exhibit two set of critical points; namely µ crit , ± = ± m (cid:96) , µ crit , ± = ± m (cid:96)m (cid:96) − , (7)where either c − or c + vanishes; see figure 1. In [14], in contrast, only the points µ crit , ± wereidentified as critical points of the theory, while nothing special was observed at µ crit , ± . In The theory also admits a Chern-Simons like formulation in terms of the vielbein e aµ and the spin connection ω abµ ; see [11, 16] for details. c − or c + vanishes, the authors pointed out that it would be interesting to see whether the logarithmicsolutions that EMG exhibits at µ crit , ± are also solutions at µ crit , ± . We answer this questionbelow. FIG. 1: µ(cid:96) (vertical axis) as a function of m(cid:96) (horizontal axis); m > µ = µ crit , ± , where c ± = 0; blue curves correspond to µ = µ crit , ± , where c ± = 0too. In the large | m(cid:96) | limit one recovers the chiral points of TMG (cid:96)µ crit , ± = ±
1. At | m(cid:96) | = 1 thecritical value (cid:96)µ crit , ± diverges and c − = c + = 0. The case m < (cid:96)µ crit , ± → ± | m(cid:96) | is large. III. EXOTIC WAVES ON BLACK HOLES
We will consider a generalization of the Ba˜nados-Teitelboim-Zanelli (BTZ) solution [21]that describes gravitational waves propagating on a stationary black hole geometry. Thisincludes AdS-waves as a particular case. We will study systematically all the possible longdistance behavior of such solutions near the AdS boundary, showing that no log-modesappear at µ crit , . 4 . Deformed BTZ geometry Let us start by considering the extremal BTZ metric with mass parameter M and spinparameter J = ∓ M (cid:96) ; namely ds M, ∓ (cid:96)M = − (cid:0) r − r (cid:1) (cid:96) r dt + (cid:96) r ( r − r ) dr + r (cid:96) (cid:18) (cid:96)dϕ ± r r dt (cid:19) , (8)where t ∈ R , ϕ ∈ [0 , π ], r ∈ R ≥ , and where r + = 4 GM (cid:96) is the black hole horizon, with G being the Newton constant. Now, perform a deformation of (8) of the type ds ≡ g µν dx µ dx ν = ds M, − (cid:96)M + h ( x + , r )( dx + ) , (9)where x ± = t ± (cid:96)ϕ . In order to solve EMG field equations, a wave profile of the form h ( x + , r ) = f ( x + ) F ( r ) must satisfy the following linear equation (cid:18) p ( r ) ddr + p ( r ) d dr + p ( r ) d dr + p ( r ) d dr (cid:19) F ( r ) = 0 , (10)where the coefficients p i ( r ) are given by p ( r ) = − r − r r + 32 r ( r − r ) µ(cid:96)r − r ( r − r )(2 r − r ) m (cid:96) r ,p ( r ) = 12 r − r r − r ( r − r ) µ(cid:96)r + 32 r ( r − r )(2 r − r ) m (cid:96) r ,p ( r ) = −
12 ( r − r ) µ(cid:96)r − ( r − r ) ( r + 3 r ) m (cid:96) r ,p ( r ) = −
12 ( r − r ) m (cid:96) r . (11)For generic values of the coefficients, such an equation has solutions of the form F ( r ) = A ( r − r ) ∆ , leading to the the indicial polynomial∆(∆ − (cid:20) ∆ − (cid:18) − m (cid:96) µ (cid:19) ∆ + 14 − m (cid:96) − m (cid:96) µ (cid:21) = 0 , (12)which, generically, has four different roots ∆ = { , , ∆ − , ∆ + } , with∆ ± = 12 − m (cid:96) µ ± m (cid:96) µ (cid:114) µ m . (13) Similarly, one may consider the extremal BTZ solution with J = + (cid:96)M deformed with a piece h ( x − , r )( dx − ) . h ( x + , r ) reads h ( x + , r ) = D ( x + ) + C ( x + )( r − r ) + B ( x + )( r − r ) ∆ + + A ( x + )( r − r ) ∆ − , (14)with A ( x + ) , B ( x + ) , C ( x + ) , D ( x + ) being four arbitrary functions that depend only on x + ;that is, ∂ − A ( x + ) = 0 , ∂ − B ( x + ) = 0 , ∂ − C ( x + ) = 0 , ∂ − D ( x + ) = 0. The constant andquadratic terms in (14), corresponding to ∆ = 0 and ∆ = 1 respectively, can be removedby local diffeomorphisms [22], i.e. they are solutions that are already present in 3D Einsteintheory. In contrast, the modes ∆ ± correspond to massive modes of EMG and are associatedto the local degrees of freedom of the theory. Solution (14) generalizes the solutions found in[14] at the chiral points µ = µ crit , ± . Geometry (9) with (13)-(14) represents a gravitationalwave on an extremal black hole. The wave co-rotates with the black hole, having an off-diagonal term g ϕt = r (cid:96) + (cid:96) h ( x + , r ) . (15)Assuming ∆ + > − <
0, this includes an asymptotically AdS solution like ds = (cid:18) r (cid:96) + (cid:96)A ( t + (cid:96)ϕ ) ( r − r ) ∆ − (cid:19) dϕ dt + . . . (16)where the ellipsis stand for the diagonal terms. This moves clockwise as the black hole. Thegeometry is wound around the horizon and the effect of the deformation h ( x + , r ) get dilutednear the boundary. The full geometry has scalar invariantsTr(Ric n ) ≡ R µ µ R µ µ . . . R µ µ n = − − n − (cid:96) n , (17)which are those of AdS space, although it is not locally equivalent to AdS . In fact, for A ( x + ) (cid:54) = 0 or B ( x + ) (cid:54) = 0, the geometry is not conformally flat.In conclusion, the propagating waves (9) can be seen as a fully backreacting, massiveexcitation of the black hole background. To reinforce this interpretation, we notice that aperturbation of the form φ ( t, ϕ, r ) = f ( x + )( r − r ) ∆ satisfies the wave equation (cid:3) ( (cid:3) − K ∆ ) φ ( t, ϕ, r ) = 0 , (18)where (cid:3) is the d’Alembert operator of the full deformed geometry (9), and where the effectivemass K ∆ is K ∆ = 4 (cid:96) ∆(∆ + 1) , (19)6hich, taking into account (13), reads K ∆ = 3 (cid:96) + m (cid:96) ( m (cid:96) − µ + 2 µ (cid:96) )2 µ (cid:96) ± (4 µ − m (cid:96) ) m (cid:96) µ (cid:96) (cid:114) µ m . (20)For generic K ∆ (cid:54) = 0, the space of solutions to the wave equation (18) is the direct sum ofthe kernels Ker( (cid:3) − K ∆ ) + Ker( (cid:3) ). At the chiral points µ = µ crit , − , K ∆ vanishes, two rootsof the indicial polynomial become zero, and a new logarithmic solution to (18) appears. Thislogarithmic solution is in the difference of kernels Ker( (cid:3) ) − Ker( (cid:3) ), as usual with confluentdifferential equations.Below, we will discuss systematically the different confluent points of the wave equationto see where logarithmic solutions actually occur. To organize the discussion, we will classifythe confluent points in terms of their degree of degeneracy of the roots of (12). The differentcases are: (a) the roots ∆ ± collide, that is ∆ + = ∆ − ; (b) one of the roots ∆ ± goes to either0 or 1; (c) the limiting case where both ∆ − and ∆ + coincide with either 0 or 1. The case (a)occurs where m = − µ and we will refer to it as the ‘degenerate point’ or the ‘confluentpoint’. The case (b) corresponds to the chiral point µ = µ crit , ± and, therefore, we will referto it as the ‘chiral point’. The case (c), to which we will refer as the ‘double confluent point’,happens when µ = µ crit , ± = ± / (2 (cid:96) ) . Finally, we will analyze the point µ = µ crit , ± , forwhich no special behavior is observed. B. Topologically massive gravity
Let us start by studying the TMG limit of the general solutions, which corresponds to m → ∞ . In this limit, the exponents reduce to ∆ + = (1 + µ(cid:96) ) / − → −∞ , yielding h ( x + , r ) = B ( x + )( r − r ) (1+ µ(cid:96) ) / , (21)together with the modes ∆ = 0 and ∆ = 1 of GR. These are a generalization of the so-called AdS-waves of TMG [3, 22]. For B = const, solution (21) is a stationary deformationof the BTZ black hole; and it is worth mentioning that this is not in contradiction withthe Birkhoff-like theorems known for TMG [23, 24]. In particular, the existence of solution(21) is consistent with a conjecture in [4], which states that, at the chiral point µ(cid:96) = − µ(cid:96) = −
1, the solution can be seen to be a7olution of GR. For µ(cid:96) < −
1, the deformation satisfies the Brown-Henneaux asymptoticboundary conditions [5] and so it represents an asymptotically AdS non-Einstein space.The case µ = 0 is special as the geometry with h ( x + , r ) = B ( x + )( r − r ) / turns outto be conformally flat without being an Einstein manifold, so it corresponds to a non-trivial solution of 3D conformal gravity exhibiting the typical linear behavior h ∼ r at largedistance. C. Confluent points
When m = − µ , the roots ∆ ± collide, i.e.∆ + = ∆ − = 12 + µ(cid:96) . (22)Since these solutions coincide, a new linearly independent solution to (10)-(11) must emerge.As probably expected, this new solution has a logarithmic behavior; more precisely h ( x + , r ) = B ( x + )( r − r ) / µ(cid:96) + A ( x + )( r − r ) / µ(cid:96) log( r − r ) . (23)Depending on whether µ(cid:96) < − / µ(cid:96) > − /
2, the function h ( x + , r ) that controls thedeformation of the BTZ geometry diverges at either the black hole horizon r = r + orat the boundary r = ∞ , respectively. At infinity, the behavior may actually be regular:The solution corresponding to the B -mode in (23) obeys the Brown-Henneaux boundarycondition if µ(cid:96) < − / D. Chiral points
More relevant for our discussion are the points µ = µ crit , ± , where one of the roots ∆ ± of the indicial polynomial degenerates to either 0 or 1 and where one of the central charges(6) vanishes. There, again, new solutions to (10)-(11) that involve logarithms appear. Suchsolutions to EMG were already studied in ref. [14]. They can be of two types:First, consider µ = µ crit , , which yields c + = 0. At this point, ∆ + = 1 and the deforma-tion (9) takes the form h ( x + , r ) = B ( x + )( r − r ) (1 − m (cid:96) ) / + A ( x + )( r − r ) log( r − r ) . (24)8or m (cid:96) ≥
1, the B -mode of (24) respects the Brown-Henneaux boundary conditions andso it gives an asymptotically AdS solution. The A -mode, in contrast, neither respect thestrong [5] nor the weakened [2] asymptotically AdS boundary conditions.Second, consider the other chiral point, namely µ = µ crit , − . In this case, ∆ − = 0 and,again, a new logarithmic mode appears. In this case, the wave profile h ( x + , r ) takes theform h ( x + , r ) = B ( x + )( r − r ) (1+ m (cid:96) ) / + A ( x + ) log( r − r ) . (25)At µ = µ crit , − , one finds c − = 0 and K ∆ = 0. In fact, the long range mode, namely thelogarithmic mode in (25), is interpreted as appearing due to the massless graviton: Thislogarithmic solution belongs to Ker( (cid:3) ) − Ker( (cid:3) ).The observation that the effective mass K ∆ of the wave equation (18) vanishes shouldnot be mistaken for statement that solution (25) has vanishing mass. In fact, it is not thecase: The A -mode in (25) respects the boundary conditions of [2] and so it can be thoughtof as a solution in AdS with non-vanishing mass. Its mass, according to the computationin [14], is given by M = 14 πG(cid:96) (cid:16) m (cid:96) (cid:17) (cid:90) π(cid:96) A ( τ ) dτ . (26) E. Double confluent points
Now, let us study the ‘double confluent points’, which correspond to µ crit , ± = ± / (2 (cid:96) ),where three roots of the indicial polynomial (12) coincide:At the point µ = µ crit , = +1 / (2 (cid:96) ), one finds ∆ + = ∆ − = 1 and c + = 0. The waveprofile h ( x + , r ) in this case takes the form h ( x + , r ) = B ( x + )( r − r ) log( r − r ) + A ( x + )( r − r ) log ( r − r ) . (27)This type of h ∼ log ( r ) solutions also appear in other higher-order generalization of TMG;see for instance [22]. Logarithmic solutions like (24)-(25) were shown to exist in TMG, NMG, MMG, and in theories such asZwei Dreibein Gravity (ZDG) [26] whenever one of the central charges of the dual CFT vanishes [27]. The B -mode, on the other hand, respects the stronger (Brown-Henneaux) AdS boundary conditionsprovided m (cid:96) ≤ − µ = µ crit , − = − / (2 (cid:96) ), on the other hand, one finds ∆ + = ∆ − = 0 and c − = 0, and,just as in the previous case, two new logarithmic modes appears; namely h ( x + , r ) = B ( x + ) log( r − r ) + A ( x + ) log ( r − r ) , (28)which, for A = 0, turns out to be asymptotically AdS in the sense of [2]. F. Are any other critical points?
Now, let us analyze the points µ crit , ± = ± m (cid:96) , where the charges (6) obtained in [16]also vanish. At those points, no logarithmic behavior near the boundary of AdS seems tooccur. Consistently, K ∆ does not vanish there. This answers the question raised in [16]about the existence of such log-solutions at µ = µ crit , ± .Actually, it is easy to see from (13) that at µ = µ crit , one has ∆ ± = (1 ± √ m (cid:96) ) / µ = µ crit , − one has ∆ ± = (3 ∓ √ m (cid:96) ) /
4, and so the solution takes thepower-like form h ( x + , r ) = B ( x + )( r − r ) ∆ + + A ( x + )( r − r ) ∆ − , (29)with no logarithmic behavior. This might seem puzzling because, as we said, in any othermassive deformation of 3D gravity that had been explored, whenever a central charge of thedual CFT vanishes log-modes were shown to appear; this happens, for example, in TMG,NMG, MMG, ZDG. This invites us to return to the question about the discrepancy between(6) and the central charges obtained in [14], the later being non-zero at µ = µ crit , ± . IV. DISCUSSION
A first observation to understand the reason for the discrepancy of the central chargescomputed in [16] and those computed in [14] is that, when taking the limit µ → ∞ in(7) and, after that, taking the limit m → ∞ , one obtains c + = − c − = 3 (cid:96)/ (2 G ), whichagrees with the central charges of the so-called Exotic Gravity (EG) [28]. In contrast, ifconsiders the result of [14] and takes the limit µ → ∞ , m → ∞ of that then one obtains c + = c − = 3 (cid:96)/ (2 G ), which is the Brown-Henneaux central charge of Einstein gravity [5].This means that the discrepancy between the charges can be traced back to the differenceof the theories that are being considered: The papers [16] and [14] are actually dealing with10ifferent theories; while [16] deals with a Chern-Simons like computation in the higher-orderextension of the EG [28], [14] deals with an Abbott-Deser-Tekin (ADT) like computation of the higher-order extension of GR. Therefore, it should not come to a surprise that thecharges do not agree. To understand this better, let us recall how it works in the caseof the undeformed theory µ = ∞ , m = ∞ : While the GR Lagrangian in terms of thevielbein 1-form e a = e aµ dx µ and the spin connection 1-form ω ab = ω c (cid:15) abc = ω abµ dx µ reads L GR = (cid:15) abc ( R ab ∧ e c + e a ∧ e b ∧ e c ) with R ab being the curvature 2-form ( (cid:96) = 1), the EGLagrangian reads L EG = (cid:15) abc ( ω ab ∧ dω c + ω a ∧ ω b ∧ ω c ) + e a T a with T a being the torsion2-form (see also [29]). Both theories yield the same field equations (i.e. the 3D cosmologicalEinstein equations) but they yield different charges. Our interpretation is that the same ishappening here with the massive deformations.Therefore, we think of the theory defined by the field equations (2)-(4) at the chiral point µ = µ crit , − as the gravity dual of a CFT with the central charges [14] c − = 0 , c + = 3 (cid:96)G (cid:16) − m (cid:96) (cid:17) . (30)At this point, the solutions of the theory exhibits the typical behavior h ∼ log( r ) of theLog-gravity [4]. This suggests that, at µ = µ crit , − , provided one considers sufficiently weakAdS boundary conditions, the dual CFT turns out to be a logarithmic CFT [30]; that is,a non-unitary CFT whose Virasoro operators L and ¯ L are not diagonalizable but form aJordan block. This means that in the CFT there exists a mixing between primary operatorsand other type of operators, called the logarithmic partners. In particular, the stress tensormay have a logarithmic partner with which it has a non-vanishing 2-point function. Thismixing of the stress tensor and its partner is controlled by a new anomaly, b , which appearsin the pole ∼ /z of the operator product expansion. In [31], a simple method to computethis anomaly for the case of a logarithmic CFT with a massive AdS gravity dual was given.Applying this method in the case of EMG, we find b = − (cid:96)G (cid:16) m (cid:96) (cid:17) . (31) In [16], it is affirmed that the main reason for the disagreement with [14] is that in the latter paper theADT method is applied in the metric formulation while the stress-tensor is not the linearized limit ofany consistent source tensor for the full EMG equations. However, since the linearized field equationsof EMG are on-shell divergenceless, their contraction with a Killing vector of the background leads to awell-defined ADT-like conserved current.
11s a consistency check, we observe that in the limit m → ∞ , µ crit , − tends to − /(cid:96) , whichis the chiral point of TMG; in that limit, b tends to − c + = − (cid:96)/G , which is in perfectagreement with the anomaly coefficient of the Log-Gravity of TMG [4, 31]. Acknowledgments
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