Volume and complexity for Warped AdS black holes

Journal of High Energy Physics, Jun 2018

Abstract We study the Complexity=Volume conjecture for Warped AdS3 black holes. We compute the spatial volume of the Einstein-Rosen bridge and we find that its growth rate is proportional to the Hawking temperature times the Bekenstein-Hawking entropy. This is consistent with expectations about computational complexity in the boundary theory.

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Volume and complexity for Warped AdS black holes

HJE Volume and complexity for Warped AdS black holes Roberto Auzzi 0 1 2 3 4 Stefano Baiguera 2 3 4 Giuseppe Nardelli 0 2 3 4 0 Dipartimento di Matematica e Fisica, Universita Cattolica del Sacro Cuore 1 INFN Sezione di Perugia , Via A. Pascoli, 06123 Perugia , Italy 2 Via Musei 41 , 25121 Brescia , Italy 3 38123 Povo (TN) , Italy 4 Piazza della Scienza 3 , 20161, Milano , Italy 5 INFN, c/o Dipartimento di Fisica, Universita di Trento We study the Complexity=Volume conjecture for Warped AdS3 black holes. We compute the spatial volume of the Einstein-Rosen bridge and we nd that its growth rate is proportional to the Hawking temperature times the Bekenstein-Hawking entropy. This is consistent with expectations about computational complexity in the boundary theory. AdS-CFT Correspondence; Black Holes 1 Introduction 2 Black holes in Warped AdS 2.1 2.2 2.3 3.1 3.2 3.3 3 Complexity=volume Einstein-Rosen Bridge Non-rotating case Rotating case 4 Conclusions A An explicit model 1 Introduction Conserved charges and thermodynamics Expectations for the asymptotic rate of growth of complexity Eddington-Finkelstein coordinates Several quantum information concepts have been fruitfully applied to the investigation of fundamental questions in gravity: classical spacetime geometry seems somehow to hiddenly encode information properties of a dual quantum system. Bekenstein-Hawking entropy is indeed proportional to the area of the event horizon [1] and the laws of Black Hole (BH) mechanics have a deep connection with thermodynamics [2]. The microscopic derivation of the BH entropy in string theory given by Strominger and Vafa [3], even if valid just for some particular extremal cases, suggests that the BH horizon area should be directly linked to the number of microstates in some appropriate dual description. The AdS/CFT correspondence provides an interesting theoretical laboratory to investigate quantum information in gravity. An example which has given us a lot of interesting insights is the Ryu-Takayanagi construction [4{6]: it links the entanglement entropy of the dual conformal eld theory (CFT) to the geometrical area of a bulk minimal surface hanging from the boundary. Entropy should be somehow related to the counting of degrees of freedom in the dual quantum description of a BH. On the other hand, it turns out that entropy is not the correct quantity to focus on in order to understand the growth of the Einstein-Rosen bridge (ERB) in the interior of a BH [7, 8]. Indeed, the ERB connecting two boundaries in an eternal AdS black hole continues to grow for a time scale which is much longer compared to the thermalization scale. Eternal BHs in AdS space are dual to two entangled copies of the same CFT living on each of the boundaries [9]. If we take both the times tL; tR of the left and the right CFTs { 1 { running forward, this geometry is dual to a time-dependent thermo eld doublet state [10]. The size of the ERB connecting the left and the right boundary asymptotically grows linearly with time. A promising candidate to capture this growth in the dual boundary theory is computational complexity [7, 8, 11{13]. For a quantum system, it is de ned as the minimal number of elementary unitary operations needed to reach a given quantum state, starting from an initial reference state. In the case of quantum mechanics with a nite number of degrees of freedom, Nielsen and collaborators [14, 15] introduced a nice geometrical tool: the problem of quantum complexity is traced back to nding geodesics in the space of unitary evolutions. In order to extend such analysis in quantum eld theories, many subtleties arise and only recently complexity calculations have been carried out for free eld theories [16{18]. Another approach to complexity in quantum eld theory uses the Liouville action [19] in connection with tensor networks [20]. Two di erent holographic duals of quantum complexity have been proposed so far: the complexity=volume (CV) conjecture [7, 8, 11] and the complexity=action (CA) conjecture [12, 13]. CV relates complexity to the volume of a codimension one surface (with maximal volume) anchored at the boundary: CV / Max V Gl ; CA = IWDW : where G is the Newton constant and l the AdS radius. CA relates complexity to the gravitational action IWDW evaluated in a Wheeler-DeWitt patch: (1.1) (1.2) Both the conjectures have their own merits. In particular, while CV explicitly depends on the AdS curvature l, CA looks more universal, because no explicit factors related to the asymptotic of the space are present. On the other hand, recent works show that CA seems to overshoot the Lloyd's bound [21] at intermediate times [22] and moreover seems to give some curious and weird features: complexity is constant before some initial time c and immediately after this time ddCA is divergent and negative. On the other hand, CV behaves as a monotonic and smooth function of the time. Another merit of CV conjecture is that it can be naturally extended to consider subregions [23{25]. Another holographic interpretation of the volume was proposed in [26]. Complexity is interesting not only to capture the linear growth of ERBs inside BHs; for example, complexity of formation of BH was studied in [27]. It is also interesting to study complexity in connection to other spacetimes; for example, the relation between complexity and spacetime singularities was investigated in [28, 29] and complexity for the AdS soliton was studied in [30]. Complexity for quenches and time-dependent couplings was studied in [31, 32]. The e ect of dilaton was discussed in [33]. It is interesting to consider extensions of holography to spacetimes that are not asymptotically AdS. The most relevant cases for physical applications would be at or de Sitter spaces; unfortunately in these cases very little is known about the dual eld theory. It is { 2 { then interesting to study non-trivial modi cations of AdS/CFT where we have more direct information about the structure of the dual. One of these cases is Warped AdS3/CFT2 [34{ 37], which is a correspondence between gravity in 2 + 1 dimensions in spaces with Warped AdS3 (WAdS) asymptotic and a putative Warped Conformal Field Theory (WCFT) in 1 + 1 dimensions. In recent times there have been signi cant progresses in the study of this extension of the AdS/CFT correspondence. Using warped conformal symmetries, an analog of Cardy formula was derived in [35]; in [36] some free examples of WCFT were built. Entanglement entropy in WAdS space and in WCFT was studied by several authors, e.g. [38{40]. In this paper we will address the CV conjecture for BHs in WAdS spaces. The paper is organized as follows. In section 2 we review the Warped black holes solurate of the volume of the ERB as a function of time, both for the non-rotating and rotating cases. We conclude in section 4. 2 Black holes in Warped AdS We will be interested in BHs with WAdS3 asymptotic [34, 41, 42]. This class of metrics should be dual to a boundary WCFT at nite temperature. For the metric we use the notation of [34]: ds2 combinations of the two mass terms [47]. We restrict our analysis to the case of positive . So at the end we will consider just the case 1. Strictly speaking, the relation between area and entropy holds just in Einstein gravity: if we consider higher order corrections to the gravitational entropy, we have to use the Wald entropy formula [48] instead of the geometrical area law. So the CV conjecture should be directly applicable just to Einstein gravity and should be appropriately modi ed in order to take into account higher order corrections in the gravitational action. A proposal for such correction has been put forward in [23, 49]. The CA conjecture can also be generalized to the case of higher derivatives corrections to the gravitational action, see e.g. [50{52]. As far as we know, there is no known non-pathological matter content in eld theory supporting stretched warped BHs in Einstein gravity [43]. However, they can be obtained as solutions to a perfect uid stress tensor with spacelike quadrivelocity [53]. Alternatively they can arise as a solution of Chern-Simons-Maxwell electrodynamics coupled to Einstein gravity [54, 55], but a wrong sign for the kinetic Maxwell term is required in order to have solutions with no closed time-like curves (which corresponds to 2 1). Moreover, warped BH can arise in string theory constructions, e.g. [56{58]. In the following we take a pragmatical approach: we suppose that a consistent realization of stretched warped BHs in Einstein gravity exists, and we investigate the CV conjecture. 2.1 Conserved charges and thermodynamics In order to compare with the expectations for complexity, we need to discuss conserved charges and thermodynamical quantities. In the Einstein case, the entropy is given by the area of the outer horizon: dM = T dS + dJ : { 4 { At least formally, we can also de ne the entropy associated to the inner horizon: l 4G S = S+ = 2 r+ is the only quantized conserved charge, we obtain J = S S+f ( ), where f ( ) is a so far arbitrary function which will be xed by thermodynamics. Imposing that the resulting dM is an exact di erential, the function f ( ) is xed and allows to solve for both the conserved charges: M = J = 16G 1 l 32G ( 2 + 3) (r + r+) 2 + 3 r r+(3 + 5 2) 2 conserved quantity associated to the Killing vector that the masses guessed by thermodynamics are indeed the same as the ones computed directly in an explicit example, which, unfortunately, has either closed time-like curves (for 2 < 1) or wrong sign Maxwell term and therefore ghosts (for 2 > 1). 2.2 Expectations for the asymptotic rate of growth of complexity In [11], it has been proposed that the asymptotic rate of increase of complexity should be proportional to the product of temperature times entropy: dC d ' T S : The main motivation comes from the fact that complexity growth rate is an extensive quantity which should have the dimensions of an energy, and which should vanish for a static object as an extremal BH. Indeed, for the WAdS BH solutions in Einstein gravity, we nd T S = (r+ r )(3 + 2 ) 16G In the next section we will nd that the growth rate of the volume of the ERB in a WAdS BH is indeed proportional to T S. The authors of [62] proposed the following bound for the complexity growth rate: dC d 2 . [(M J Q)+ (M Q) ] ; where indicate that the corresponding values of the quantities are computed at the outer and inner horizons. With suitable units for complexity, the bound (2.14) seems to be saturated in several cases. For WAdS BHs, the angular velocities computed on the inner and outer horizons are: { 5 { : J + = l 2 r+ If we use the values of mass and angular momentum in eqs. (2.10){(2.11), we nd that For the purpose of the case studied in this paper, the saturation of the bound in eq. (2.14) is equivalent to eq. (2.12). The Eddington-Finkelstein (EF) coordinates can be introduced using the light-like geodesics of the metric in eq. (2.1). A system of EF coordinates for the WAdS BH was already introduced in [61]. The coordinates that we introduce here are not the same, but they are still a system of non-singular coordinates at the horizon, de ned using infalling lightlike geodesics, that we nd convenient for our purposes. We have the following conserved quantities along geodesics: K = 2t_ + 2 r pr+r ( 2 + 3) _ ; P = 2 r pr+r ( 2 + 3) t_ + (r) ; where dots denote derivatives with respect to the geodesic a ne parameter. The null geodesics are found by imposing the additional constraint ds2 = 0. Solving the equation of motion and specializing to K = 0, we get a particular set of geodesics satisfying t_ = P 2 r and allow to de ne EF coordinates (u; u) such that du = dt + 2 r The nite expression for the coordinate change is u = t + r (r) ; u = where r (r) = 2 r+ r+ ; r In terms of these coordinates, the metric becomes ds2 l 2 = du2 drd u + 2 r pr+r ( 2 + 3) dud u + r 4 (r)d u2 : { 6 { 3.1 Einstein-Rosen Bridge Kruskal extension for WAdS BHs was studied in [61]. The Penrose diagrams for WAdS BHs are the same as the ones for asymptotically at BHs in 3 + 1 dimensional spacetime: for the special cases r 2+3 , the diagram is the same as the one for the Schwarzschild BH, while for generic rr+ it is identical to the one for the Reissner-Nordstrom BH (see gures 7 and 8 of [61]). It is important to emphasize that in the = 1 case, which is the AdS case, the Penrose diagram is di erent and is the usual AdS one. As done in [8, 11] for the AdS and the at cases, we consider an extremal codimension one bulk surface extending between the left and the right side of the Kruskal diagram; we denote the times at the left and right sides as tL; tR, respectively. The dual thermo eld double state has the following form: HJEP06(218)3 j T F Di / X e En =2 iEn(tL+tR)jEniRjEniL ; n where jEniL;R refer to the energy eigenstates of left and right boundary theories, inverse temperature. The usual time translation symmetry in Schwarzschild coordinates corresponds to a forward time translation on the right side and a backward translation on the left one [9], i.e. tL ! tL + t ; tR ! tR t : This corresponds to the invariance of the thermo eld double state under the evolution described by the Hamiltonian H = HL HR in the associated couple of entangled WCFTs. If instead we take time running forward on both the copies of the boundaries, we introduce some genuine time dependence in the problem [10] and the volume of the maximal slice will depend on time [11]. We will then consider the symmetric case with equal boundary times tL = tR = tb=2 : 3.2 Non-rotating case In this section we will compute the volume of the ERB as a function of time [11]. We rst study the non-rotating case, setting r+ = r0 and r = 0 in the metric in the coordinates (2.23). The minimal volume is chosen along the 0 u 2 coordinate, and with pro le functions u( ), r( ), written in terms of some parameter . The volume integral will run from some min to some max, with associated radii rmin and rmax: V = 2 2 Z max min d l 2 s u_ 2r 4 [3( 2 1)r + ( 2 + 3)r0] u_ r d L(r; r_; u_ ) : (3.4) The factor 2 takes into account the two sides of the Kruskal extension, the 2 is the result of the integration in u and the dots denote derivatives with respect to . The radius rmax plays the role of an ultraviolet cuto ; we will take the limit rmax ! 1 at the end of the { 7 { (3.1) is the (3.2) (3.3) = 2:5 and r0 = 1. The E = E0 line, which sits at constant rmin = r20 , corresponds to the large tb limit. Penrose diagram coordinates from [61] have been used. calculation. The conserved quantity from translational invariance in u gives It is then useful to gauge the parametrization symmetry for in such a way that V = 1 r + We can then solve for r_; u_ : r_ = 2 The minimum radius rmin is a solution of r_ = 0: 2 rmin r0rmin + 4E2 (3 + 2) = 0 ; where the physical solution relevant for holographic complexity is the one with the + sign. Conventionally, tb = 0 corresponds to E = 0 and rmin = r0. The tb ! 1 limit, instead, corresponds to coincident roots for rmin in eq. (3.8), i.e. E ! 4 The minimal value of the radial coordinate is inside the black hole horizon r20 r0 p 2 + 3 and rmin = r20 . rmin r0. The volume can be obtained as an integral in dr: 4 l2 R d : u_ 2r 4 3 gure 1. 2 s V = 4 l2 = 2 l2 Z dr r_ Z rmax s rmin (3.5) rr_ 2 : (3.6) (3.7) (3.8) (3.9) The di erence of u coordinates is: rmin dr u_ r_ r = rmin dr " 2 Note that this integral is not divergent for r ! r0. The volume can then be written as follows: V 4 l2 = E(u(rmax) u(rmin)) + rmin dr is nite and can be identi ed with the time at the right boundary. In the limit rmax ! 1, we can use the explicit expression obtained specializing eq. (2.21): u(rmax) = tR + r (rmax) ; u(rmin) = r (rmin) ; because t = 0 at r = rmin by symmetry considerations. Taking into account that both E and rmin depend on tR (see eq. (3.8) for the relation among rmin and E), the time derivative of eq. (3.11) gives, after several cancellations among terms: 1 dV 2l dtR = dV d Computing the constant of motion E in eq. (3.5) for the particular value r = rmin shows that E > 0 for > 0 (corresponding to u_ > 0) and E < 0 for < 0 (corresponding to u_ < 0). Numerical calculations with the full time dependence can be obtained by expressing in terms of E using eqs. (3.10){(3.13), are shown in gure 2. For = 1 the results in [11, 22] are recovered, under the change of variables in eq. (2.4). 3.3 Rotating case We use the metric in the coordinates (2.23); the volume functional is: V = 4 Z max min d l 2 s ru_ 2 4 u_ 2 2 r { 9 { V 1.5 0.5 0 0 r_ = 2u v u t u_ = 2 6 6 4 r ru_ 2 4 4 0.1 Due to the axial symmetry, the volume is taken along the u direction. As before, we nd the conserved quantity: 2r_ 21 2 r pr+r ( 2 + 3) u2_ 2 r 2 The expression greatly simpli es choosing a parametrization for which corresponds to setting such that V = 4 l2 R d , u_ 2 2 r 2 = 1 : This gives: Solving eqs. (3.18), (3.19), we obtain the expressions: E = u_ (r r )(r r+) + 2 r As in the non-rotating case, t = 0 at r = rmin, and so: By direct computation, we nd the relation r )(r r+) The minimum value rmin of the radial coordinate is obtained by solving r_ = 0: r )(r r+) # r )(r r+) E 2 r pr+r ( 2 + 3) # r )(r r+) + E(u(rmax) u(rmin)) : (3.26) The tb ! 1 limit corresponds to E ! (r+ 4 r ) p 2 + 3 and rmin = r++2 r . As in the non-rotating case, the physical solution relevant for holographic complexity is the one with the + sign. Conventionally, tb = 0 corresponds to E = 0 and rmin = r+ + r . The volume can be expressed as an integral in dr as: V = 2 l2 r )(r r+) r )(r r+) It is useful to introduce the di erence among the extremal values of EF coordinates: Using the previous de nitions and simplifying the expression, we obtain again the result where = l tb = 2l tR. At large , E approaches the constant dV d (r+ E0 = lim !1 d dV Numerical calculation are shown in gure 3. As a consistency check, putting = 1 for the BTZ case, we nd 3.5 2.5 3 2 1 0 which is the same result found in standard coordinates on the Poincare patch when we perform the change of variables (2.4). The late time limit of the maximal volume slices can be found also in a simpler way, as in [11]. In this limit, we expect that the maximal volume slice sits at constant r, due to translation invariance in time. We can then consider volume slices at a constant r = r^. Extremizing the volume from the metric in eq. (2.1), we nd that the only possible maximal constant-r slice sits at r^ = Inserting this value back in the volume functional, we recover eq. (3.27) with E = E0. 4 Conclusions The result of our calculation gives that the volume of the extremal slices in WAdS is a monotonically growing function for > 0, whose late time growth rate approaches to dV d ! 2 l (r+ p r ) 3 + 2 = ST p 8 Gl 3 + 2 ; where S is the Bekenstein-Hawking entropy and T the Hawking temperature. The late time rate vanishes for extremal black holes (r+ = r ) and is proportional to T S. In AdSD, we have that the coe cient of proportionality between complexity and volume [8] is usually taken as: The late-time rate of growth of the volume is: C = (D 1) V Gl : lim !1 d dV = For comparison, in the case of at spacetime BHs, lim !1 d dV D Grh ST ; 3 (4.4) (4.5) the asymptotic complexity increase rate would be still T S for every . It would be interesting to nd arguments in order to be able to discriminate among these various possibilities. From general considerations based on limits about the speed of computation, there is a conjectured bound on the growth rate of complexity of a physical system [21]. This is called Lloyd's bound and states that where E is the energy associated to the physical system and units ~ = 1 are used. In the case of BHs, the energy is identi ed with the BH mass, E = M . In general relativity the de nition of mass M depends on the choice of the asymptotic Killing vector used to de ne the conserved quantity. In the WAdS case, the boundary spatial direction in the metric (2.1) is selected by the r2 divergence at r ! 1 in the d 2 coe cient of the metric. is natural, because the norm of t does not diverge at the boundary. If we would choose another Killing vector to de ne the mass case ( then, for non-zero j, the norm of j would diverge as r2 for r ! 1. Instead in the AdS = 1) the r2 divergence of the metric at in nity disappears and t is not a natural time Killing vector, because it is not an eigenvector of the boundary metric. where rh is the horizon radius ( refer to a neglected order one prefactor [8]). Consequently, the proportionality coe cient between the late time rate of growth of the volume and T S depends on the kind of asymptotic of the spacetime. In order to compare with the AdS3 case, we can write the rate of growth of the volume in WAdS as: dV d ! ST 4 Gl ; = p 2 3 + 2 We may interpret the details of this result in distinct ways, depending on the exact holographic dictionary that we may conjecture between volume and complexity. For example, it could be that complexity approaches at late time to T S (note that 1 if we impose 2 1); if this is true, warping would make complexity rate decreases. On the other hand, it could also be that in spaces with WAdS asymptotic the holographic dictionary between complexity and volume is changed by some non-trivial function of the warping parameter ; for example, if we would have that C = V ; 2 Gl dC d 2E ; j = t j Since the late-time complexity rate is proportional to T S, if we want that an universal Lloyd's bound holds, we must require that some positive constant k exists such that kM T S for every value of r+, r at xed . If we impose this, we nd that k g( ), where g( ) = p which is a decreasing function of , with g(1) ! with appropriate normalization, to introduce a Lloyd's bound proportional to M for = 1 (the AdS case) instead this is not possible; but indeed we know that in this case AdS; in this case there is a conjectured Lloyd's bound in terms of the usual time direction in AdS [13]. For < 1 we do not expect a Lloyd's bound, because there are closed time-like curves in the geometry; in this range the mass M can even be negative. Several problems are left for further investigation: In this note we considered the case in which WAdS BHs are realized as solutions of Einstein gravity with some appropriate matter content. These objects can also be realized as vacuum solution of TMG and NMG. In these cases we expect some higher order corrections to the CV conjecture, analog to the area corrections in the Wald entropy formula. Some proposals have been discussed in [23, 49]. It would be interesting to compute these corrections explicitly. The CA conjecture should also be investigated for the WAdS BH solutions, both in the asymptotic rate of growth and in the initial transient period. This was initiated in [51] for the case of TMG; in this case the late-time complexity rate is not proportional to T S, but it still vanishes in the extremal case. It would be interesting to study complexity in the boundary Warped CFTs. Lagrangian examples of free Warped CFT were introduced in [36, 37]. Acknowledgments A An explicit model We are grateful to Shira Chapman for useful comments on the manuscript. In this appendix we consider an explicit model whose entropy satis es the area law and admitting the metric eq. (2.1) as a solution [54]. This is a model of Einstein gravity in (2+1) dimensions minimally coupled to a gauge eld with Chern-Simons and Maxwell terms: S = 1 is the Levi-Civita tensorial density. Here we put a coe cient of the Maxwell kinetic term. The equations of motion for the gauge eld are ; = (A.1) (A.2) D F = pg F ; while the Einstein equations are G 1 L2 g = 2 T ; T = F F g F F We consider the set of coordinates (r; t; ) where the metric assumes the form (2.1), and we choose a gauge motivated by the ansatz from [54]: A = adt + (b + cr)d ; F = c dr ^ d ; where fa; b; cg is a set of constants. In this gauge, the Maxwell equations give: = l From the Einstein equations, in order to require absence of closed time-like curves ( > 1), we have to choose = 1 and the following value of the parameters: L = l r 3 2 2 ; r 3 2 c = l ( 2 1) ; which are simultaneously de ned only when 1 < 2 < 3. So there is con ict between absence of closed time-like curves and presence of ghosts ( = 1). In ref. [54] the conserved charges associated to asympthotic isometries of the black hole have been computed starting from the following form of the metric in the coordids2 = pdt~2 + + 2hdt~d~ + qd~2 ; nates (t~; r~; ~): with functions given by and U(1) gauge eld where where and is a gauge constant. 2 L2 ; p(r~) = 8G ; q(r~) = h(r~) = 2 r~ ; (A.8) A = At~dt~+ A~d~; At~(r~) = 4G + ; A~(r~) = Q + r~ ; (A.9) 2 L We can put the metric (2.1) in the form (A.7) by means of the coordinate change: t~ = r l3 ! t ; r~ = r ; ~ = p !l3 2 ; ! = 2 + 3 (A.3) (A.4) (A.5) (A.6) (A.7) (A.10) (A.11) (A.12) HJEP06(218)3 The previous set of transformations is such that the gauge eld in the coordinates (t; r; ) can be written as A = adt + (b + cr)d ; motivating the ansatz (A.4). The quantities ; J ; Q appearing in the previous solution are respectively identi ed with the mass, angular momentum and charge of the black hole. The equations of motion and the change of coordinates do not uniquely x the charge Q, while we identify = J = 2 + 3 16Gl2 As it is pointed out in [54], the set f ; J ; Qg satis es the rst law of thermodynamics in the form d = T dS + dJ + totdQ ; where the total electric potential is shown to be tot = 0; thus eliminating the contribution from the charge of the black hole. This special form of the rst law of thermodynamics is a consequence of the choice of the Killing vectors associated to mass and angular momentum in [54], since all the contributions coming from the charge are eliminated. A direct match with the mass M and angular momentum J coming from the thermodynamic analysis in (t; r; ) coordinates gives: = M l In order to get the conserved charges associated to isometries in (t; r; ) coordinates, we need to adjust the normalization conditions: The angular range 0 2 corresponds to 0 such as mass, entropy and angular momentum in (t; r; ) coordinates get an extra factor if we want to preserve the length of the integration along [0; 2 ]: ~ 2 p!2l3 , so extensive quantities, Killing vectors are transformed as: 2 In [54] it is de ned = h(r+)=q(r+); while in eq. (2.10), (2.11) we followed the conventions of [34], where an additional factor of l is put in the denominator both for the angular velocity and the Hawking temperature. 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Roberto Auzzi, Stefano Baiguera, Giuseppe Nardelli. Volume and complexity for Warped AdS black holes, Journal of High Energy Physics, 2018, 63, DOI: 10.1007/JHEP06(2018)063