JHEP03(2017)119 Springer March 14, 2017 March 23, 2017 February 6, 2017 : : : November 14, 2016 : , Revised Accepted Published 10.1007/JHEP03(2017)119 Received doi: Published for SISSA by [email protected] , . 3 1610.02038 The Authors. c AdS-CFT Correspondence, Black Holes, Gauge-gravity correspondence, Mod-

In this paper, we study the physical significance of the thermodynamic volumes , [email protected] E-mail: [email protected] Theory Group, Department of PhysicsUniversity and of Texas Texas Cosmology at Center, Austin, 2515 Speedway, C1600, Austin, TX 78712-1192, U.S.A. Open Access Article funded by SCOAP ArXiv ePrint: bound whereas CA-duality is not. Keywords: els of Quantum Gravity hole. We go on toof discuss black the hole role of solutions, thermodynamicswe and in dub then complexity point “complexity = out action =thought for the volume a of possibility 2.0”. number of as an In theevidence alternate this spacetime that, proposal, alternate volume in which proposal of certain the cases, the complexity our Wheeler-DeWitt would proposal patch. be for Finally, complexity we is consistent provide with the Lloyd the extended thermodynamics interactsBrown with et al. the (CA-duality). recent We, complexityrate in of = particular, change of action discover complexity that has proposal theirreminiscent a of proposal nice of decomposition for the in the terms Smarr late of relation.via time thermodynamic CA-duality quantities This holographic, decomposition interpretation for strongly the suggests thermodynamic a volume geometric, of and an AdS black Abstract: of AdS black holesthis using formalism the to Noether study charge the formalism extended of thermodynamics Iyer of and a Wald. few After examples, applying we discuss how Josiah Couch, Willy Fischler and Phuc H. Nguyen Noether charge, black hole volume, and complexity JHEP03(2017)119 ], do (1.1) 4 of the – 1 H 7 10 17 5 13 ], and more recently high energy 9 – 5 . S 22  22 ∂P ∂H 25  – 1 – 3 . The volume, then, can be defined in the usual = 18 U 24 V 15 11 19 21 1 25 ], have suggested that the pressure should be identified as the cosmological 10 5.4 Altering the bound by a pre-factor 4.2 Rotating black hole 5.1 The Lloyd5.2 bound with conserved Bound charge violation:5.3 near the Bound ground violation: state exact results 3.2 CA-duality, through3.3 the lens of Complexity black = hole volume chemistry 2.0 4.1 Electrically charged black holes 2.1 Application to2.2 Einstein-Maxwell: charged BTZ Application black to hole Einstein-Maxwell-dilaton: the R-charged black hole 3.1 Review of Brown et al. constant. In this framework,hole chemistry”, dubbed the the ADM extended masssystem black of rather the hole black than thermodynamics hole its or isthermodynamical internal reinterpreted “black way energy as to the be: enthalpy not include a pressure-volume conjugateto pair. the This difficulty of conspicuous defining absencethe the is volume perhaps area of related of a black thethe hole horizon, in foliation a a of coordinate-invariant integration way: na¨ıve spacetime. overphysicists unlike the [ A interior number of of a relativists black [ hole depends on 1 Introduction The laws of black hole thermodynamics, at least in their traditional formulation [ 6 Conclusion 5 Action or volume? 4 Thermodynamic volume and complexity: conserved charges 3 Thermodynamic volume and complexity: Schwarzschild-AdS Contents 1 Introduction 2 Volume and Iyer-Wald formalism JHEP03(2017)119 ] ]. 31 26 (1.2) , 25 ]. 17 – 11 ] and the fidelity [ 30 – 27 . 2 2 Ω ] in conjunction with the kinematic 24 gdrd − ]. √ + ] to relate matter in the bulk to the relative r 26 , 0 21 Z ] to relate quantum information inequalities to 25 – 2 – = 23 3 + πr 4 3 ], or a slight twist thereof. This is the main finding of ] to relate canonical energy in the bulk to the quantum = 22 19 V , 18 ), we will relate the thermodynamic volume (and also the Smarr relation) ] where the formalism was used to derive the linearized equation of motion 4 20 and . A powerful way to derive the first law of black hole thermodynamics, the Iyer- 3 2 ). For a selection of work on or related to this topic, we refer to [ The second reason to suspect that the thermodynamic volume is relevant to holography The first reason to believe that the thermodynamic volume has a place in holography Our main goal in this paper is two-fold: on one hand, we attempt to shed light on In more complicated cases, such as rotating holes or solutions with hair, the thermo- 1.2 Ryu-Takayanagi surface and the boundary.the In thermodynamical the volume light of also these admitsand ideas, a we it quantum will is information suggestive find theoretic(sections that interpretation, in this paperto that the it complexity indeed as per seems the so. proposals in In [ the second part of the paper is that various notions of volumetheoretic in quantities. the bulk In have particular, been the identifiedAdS size with black of quantum hole information the is Einstein-Rosen believed to bridgeFurthermore, capture of the the a complexity complexity of 2-sided of subregions eternal the of thermofield thehave double boundary been state CFT [ [ related to the volume of a constant time slice in the bulk enclosed between the more examples, the formalism wasentropy used on in the [ boundary,Fisher in information [ on thegravitational boundary, positive in energy [ theorems, and finallyspace in program [ to clarify the emergence of gravity from entanglement. natural step to extend the formalismthe to Iyer-Wald formalism derive has the proved thermodynamic useful volume.the to In holographers geometry recent as in years, a means thestarting to bulk with translate [ between to quantumin information the theoretic bulk from quantities the on first the law of boundary, entanglement on the boundary in pure AdS. To give a few is that, as we(or will Iyer-Wald) demonstrate formalism below, [ thissection quantity is derivable fromWald the formalism Noether has charge yielded deep insights into the nature of black hole entropy, so it is a abstract notion of volume associated tovolume the of black hole any and spatial does region. notto On correspond holography, the to and the other actual hand, in we particularshould will to relate we the the believe thermodynamic quantum that volume context complexity the ? of thermodynamic To the this volume question, boundary we has state. will a offer Why four role answers, which to we play list one in after another the below. holographic dynamical volume is less intuitive,arises and as it an is integral an ofto interesting some ( question local to quantity ask over some how the region of volume spacetime, inthe a meaning way of similar the thermodynamic volume as a geometrical quantity, which, a priori, is an Nordstrom (AdS-RN) black hole,gration the over thermodynamic the volume “black coincides hole with interior”: a inte- na¨ıve Fascinatingly enough, in simple cases such as the AdS-Schwarzschild or AdS-Reissner- JHEP03(2017)119 , ] 4 35 , 19 , 18 ], the black hole 32 ] and dubbed “complex- 34 , we contrast our proposal for the 5 , we briefly review the Iyer-Wald for- 2 , we conclude and discuss future work. – 3 – 6 ]. The first way, first proposed in [ 34 , , together with a solution with a bifurcate timelike Killing vector  33 L , R 26 , = 25 L R = , we move on to discuss the connection between the extended thermodynamics S 3 ] and dubbed “complexity=action” or CA-duality, postulates that the complexity is The paper is organized as follows: in section However this linear growth can be observed for various geometrical entities crossing The third motivation to study the thermodynamic volume in holography is the question 25 2 Volume and Iyer-Wald formalism In this section we will presentwhich a will allow slight us generalization to of derive the theaction Iyer-Wald volume. formalism The [ formalism requires a diffeomorphism invariant electrically charged and rotatingcomplexity against holes). CA-duality, and In showproblems that, section which in ail certain CA-duality. In cases, section our proposal can help fix malism (with varying cosmologicalvolume constant) of and two apply solutions:section it the to charged derive BTZand the black the thermodynamic hole complexity and inwe the the extend R-charged simple this case black connection of hole. with the the complexity In AdS-Schwarzschild to black a hole. black hole In with section conserved charges (i.e. WDW patch at late time.= Volume We 2.0” will in also whichpatch point complexity out rather is a than identified third with thediscussed possibility, the dubbed in action. spacetime “Complexity more volume detail This of further is the in the WDW potentially paper. even easier to work with and will be in [ dual to the bulksolves action the evaluated lengthscale on problem the ofthat Wheeler-DeWitt the CV-duality, and WDW (WDW) has patch patch. is inpaper easier addition This that to the proposal work the practical with advantage thermodynamic than the volume maximal is volume. intimately We related will see to in the this linear growth of the of an ERB and capturethe the literature complexity ? [ Two waysity=volume” to or achieve this CV-duality, have postulates beenmaximal that considered spatial in the slice complexity crossing isat the dual late ERB. to time, This has the proposal,for the while volume which minor capturing of there problem the the that linear seems growth a to length be scale no has unique, to be natural introduced choice. “by hand”, The second way, first proposed observation is that the sizea of the behaviour ERB conjectured grows to linearly be with true the boundary of time the at quantum latethe complexity time, at wormhole, exponentially and late to time. leads correctly us pick to the out fourth one and among last them motivation of is this a paper: non-trivial how problem. to correctly This quantify the size of to what extentinterior holography knows was about probed the“vertical with black entanglement” hole minimal interior. as surfaces well Informulated. which as [ cross a Like tensor the theuseful network horizon, way quantum picture to and complexity, of keep the the track black of notion hole vertical time interiors of entanglement in were could the dual serve CFT. as In a the case of the complexity, the key JHEP03(2017)119 (2.1) (2.6) (2.7) (2.8) (2.9) (2.2) (2.3) (2.4) (2.5) runs over the φ , and considering some . ξ 2 1 − − d d a a dx dx , d ∧ a ∧ Λ φ. δ . dx ζ ... ) ) ... δ L Λ ∧ δφ φ L ∧ φ ∂ ∂ ∧ ) µ · δφ . 1 L E 1 ( a φ ... ζ . a Λ ∇ ∂ + Consider some general variation of the µ ) ( δ ∧ Θ L dx φ ζ dx − · 2 ∇ 1 ∂ 1 ∂ X ( L Λ 1 δφ ( a − ) dimensions: ]. ξ − µ d ∂ ∂ φ , and where the sum over d Q φ ∂ dx d · φ ζ d E 20 − ) + d ...a , this becomes δ ...a ξ 1 ) φ ( φ ζ 1 − ∇ φ ...a ξ ζ – 4 – X − 1 ( µa X δ Θ ) = µνa a L ( ζ = Q ∂φ ∂ ( + g ε g ε Θ φ g ε ) = by the killing vector field J χ − δ ) = d − − Θ = d ζ √ δφ ζ √ d ( √ = ( = φ ! J 1 d 1)! 2)! E = Θ χ L 1 ) such that on-shell ζ 1 − − = L δ ζ d d δ  ( ( ( Q = = , we now define µ  µν δφ  denotes the usual volume form in  . We will follow the notation used in [ ξ In particular, 1 We will find it useful to define the additional forms: general other variation By an algebraic computation we may see that on-shell and a Noether charge form Replacing our general vector field In this case, since ourto action derive is the diffeomorphism conserved invariant, we current may apply Noether’s theorem is the equation ofentire motion field form content for ofdiffeomorphism the the generated field by theory. a In vector field the case where this variation is due to applying a is the symplectic potential current, Lagrangian. For an action includingwrites: a cosmological constant which is allowed to vary, one where field JHEP03(2017)119 (2.12) (2.11) (2.10) ) and inside + i λν r > r F λ ξ ) + . λ χ . A H  2 λ  2 2 q Z ξ r 2 ( L F ν dφ − L  1 4 2 ∇ r χ −  − log
+ ∞ 2 r L µν Z ) q r 2 r F 2 2Λ) 1 2 ( = µν dr µ  f µ − − dt. log −  πL  L Λ  µν µ 2  µ + 2 R ∂ ∂ 2 16 q λ ( r L r J Q i · respectively. For this choice of Killing vector 2 L 2 ξ  λ . – 5 –  ξ L dt g = Θ = = − νλ . +  A 2 ) Σ r λ λ − r R J Z log Q ( Θ ξ m − A . Both outside the outer horizon ( √ = f 2 q µν Λ λ 2 µν x − V dP g δ ξ r − − − r d r 4 F d − + µν = = = ] − is a bifurcate time-like Killing vector field, whose killing = F = ) = µ Z 2 ] r A and r 1 2 ξ t µ = 1. This solution has two horizons, an outer horizon at tr ( χ ν ds ∂ = ξ [ + d f ν − Q [ r G ∇ S ] ), Σ ν µ ∇ Z − ξ = 2 ∇ ν h [  r ν ∇ 2  h ∇ r < r = = = µ µ µν J Θ Q and an inner horizon at + r = the inner horizon ( horizons are given by field, we find that the Noether charge only has one nonzero component: Here we use unitsr where 4 Let us now specialize tohole a in solution of 3 the dimensions. Einstein-Maxwell The system: metric the together charged BTZ with black the gauge field are given by: with After some algebra, we findcharge the to symplectic be: potential current, Noether current and Noether from the left above gives rise to the 2.1 Application toNext, Einstein-Maxwell: we charged apply BTZ the black above formalism hole to the Einstein-Maxwell theory: the bifurcate killing horizon and the conformal boundary at infinity then yields In the case ofthermodynamics a upon black evaluation hole of the spacetime, integrals, this where roughly reduces speaking, to the the second term extended first law of black hole Applying Stokes’ theorem to this form on a region Σ of a constant time slice bounded by JHEP03(2017)119 R (2.19) (2.13) (2.14) (2.15) (2.16) (2.21) (2.18) instead cutoff r δP after some algebra. . δL  2 2 q δL R T δS 3 3 1 4 2 − − L r L − 2 + r = 2 2 r L 4 3 = yields  L − = 0 (2.20) = 0 (2.17) 4 3 r g , but putting in a large L L δL −    − δL 2 2 2 g δP q
√ q = and L δL. − 2 . Rewriting this in terms of 2  1 4 2 → ∞ 2 q t 2 dr q L √ q 2 q dφ r r − 0 2 1 4 1 4 r − 2 L 3 r dr 3 T δS L 2 vanish. For a perturbation defined by per- L 2 − Z − − µνφ R − + r L 4 1 πL ε − for this horizon, and evaluating at the singu- r πL 2 2 2 − + µν  Z 8 dφ 2 r r r − µν 16 – 6 – Q r 4 π   4 2 2 + L dφ δL we can recongnize this as gχ 0 r T δS π π 3 π Z − =  2 + + diverges as = = 0 r = π √ π tr Z − 4 δL r χ − + π ) χ + 2 fixed then becomes π V V Θ 0 = 4 δL q Z we read off the volume for the inner horizon: T δS L Λ = ( T δS = δ δ · and L Λ χ δP δ δ ξ of the solution above and leaving the other parameters fixed, we · m L Z for ξ once again yields Λ =const Z δ r δL − Z . On the other hand, Λ r is to emphasize that the quantity enclosed pertains to the inner horizon. δ δL = − 2 L ) r q 4 − ... ) at 2.15 , the first law with δL to be: where the ( Once again trading All in all, we obtain the first law: We could equally well havetion done ( this in regionlarity inside gives the inner horizon. Evaluating equa- and so we have the volume we get a cutoff independentof result also equal to Evaluating this on the outerOn horizon the other hand, theand integral adding of χ Integrating this form over any surface of constant turbing the AdS length find that with the other two components zero. From these we find the only nonzero component of and all other independent components of JHEP03(2017)119 (2.33) (2.34) (2.26) (2.27) (2.28) (2.31) (2.24) (2.29) (2.22) (2.23) (2.25) 1) , ε 1 − ! ,  )) 2 1 i Ω − φ ( d 2 r = ( − V 4 + a 2 2 ) i f dr 1), ) (2.32) ∂φ i + −  , r (( φ 2 ( 1 / 2 1 , 3 I =1 / ) (2.30) i 1 X 1 H I ) cosh m − − I q π I ) m 1 4 J =1 H i I q + Y 32 I q = ( + 2 X H 4 + 4 =1 + 2 I 3 =1 + − 2 π Y r + I I q J X a I g ) 2 4 ( Y ) dt q I √ + – 7 – 4 1 2 I ( 2 ( ) r + − q I r π F 1), r 4 fdt =1 q + 2 m ( 2( ~ 4 φ Y 2 I p 0 · − g 4 / I , 1 2 f p / I ) = 1 m π ~a i 1 1 q J + + 2 − e I φ = = = = = − I H ( I + H , q m 4 r J I I =1 r V ( S T r H q 2 I X M I =1 Φ 4 Q =1 q 4 J Y I π − 1 = (1 8 p Q − 2 − a = 1 + = = = = 1 π 2 I ~ φ J f · R 1), I A 16 H , ds ~a 1 1 2

, ]. In (3+1) dimension, the action is given by: − e = 36 L = (1 1 a The thermodynamical quantities are: The metric together with the matter fields are given by: with and In this subsection, we considerR-charged a black more hole complicated in example, 4been and dimensions. studied derive in The the [ volume thermodynamics of of the R-charged black holes has 2.2 Application to Einstein-Maxwell-dilaton: the R-charged black hole JHEP03(2017)119 is = 0 1 r q (2.38) (2.35) (2.36) (2.40) (2.37) ). The ). 1.1 1.1 ), and r is the largest 0 r 1 − ) = 0 (2.39) + r relabeled to K ( q K + J in order to recover the usual r q H 2 2 I q q Ω =1 4 as a partial derivative ( − 2 2 X K r Ω ) V X gdrd + ) (with 3 → r Imass is volume now can reinterpreted be as computed the using enthalpy and the familiar thermodynamic formula: In the extended phase space, the pressure is the cosmological constant, which is also the JHEP03(2017)119 χ ) cb and ) I (2.47) (2.45) (2.46) (2.42) (2.43) (2.41) ( δQ F c , we find: ξ + r evaluated on ) + 2 c  ) χ I c δg ( 2 r # A c δg K ξ q  ( b 4 J ∂ q H ( 2 2 I ) 3 q I Ω ab term with the integral of ( H 2 F ~ φ · H I X gdrd 1 T δS ~a I

χ ∂ ∂g 2 ∞ Z δg Z 2 = = of the black hole rather than its ) or the central charge on the field theory side, and that the volume can δg ] that varying the cosmological constant in the bulk corresponds to varying N 37 ) and the horizon term (the lower limit of integration) in ( , V dP . Moreover, it remains unclear as to what the information contained in the 16 2.46 , exterior 15 , interior In this section, we bring the two questions above together (the black hole interior and We also note here that there exists an alternative approach in the literature to derive the volume, which 13 2 , complexity of the CFT state is dual tois based the on integral theKomar of Komar formula formula, the as see bulk a e.g.potential. integral action [ over over the the exterior so-called of the black hole, and the integrand is basically the Killing be thought of as acounted chemical potential-like by quantity the corresponding central to charge. the degrees of freedom the CFT interpretation) andof attempt quantum states. to answer In them a through series of the elegant notion papers of [ begs the question of whether thehole thermodynamic volume has something to dovolume with can the teach black us about the11 dual CFT. It isthe generally rank pointed of out in SU( the literature [ regularized by the Iyer-Wald form 3 Thermodynamic volume andFrom complexity: the viewpoint Schwarzschild-AdS ofvolume arises the as Iyer-Wald an formalism, integral over as the we have seen above, the black hole extended Iyer-Wald formalism, thefinite lower term limit in of the integration us integral that of the volume ofover the the black hole isarises perhaps as best thought the of integral as of arising the from an scalar integral potential over the whole black hole exterior, but it is and we recover equation ( If we now compare thepairwise, divergent and terms in we ( areterm left in with ( aobtain: finite answer which consists of two parts: (1) the finite here that the extended Iyer-Waldapproach formalism makes described this in fact manifest,will equation in diverge ( contrast and with we the then have find: to regularize by a radial cutoff Notice that we have an integral of the scalar potential on the right-hand side! We emphasize JHEP03(2017)119 3 (3.1) R t i ]. The regularization introduces 38 . From these two points on the R t , rather than the action, of the is equivalent to choosing a quan- TFD | . Alternatively, one could move the two ) R + R t r t R  H and + L , as done in [ t L t L cutoff r H B' ( cutoff i r B – 11 – − e spacetime volume = i ) R , t L t ( , the patch loses a sliver and gains another one (depicted in darker ψ | L δt L + L t for a depiction), we draw four null rays, and the WdW patch is the +δt L L t t 1 and the time on the right boundary as L t is shifted to . The Wheeler-DeWitt patch of the AdS-Schwarzschild black hole (depicted in orange). L t On the CFT side, picking out two times The WdW patch as described here extends all the way to the boundary, and therefore the action 3 evaluated on the WdWcould is simply divergent. cut To off extractcorners the a of patch finite the at answer,terms WdW some we which have large patch drop to radius out on choose whenregularization the a we unspecified. take regularization. boundary the to One time derivative of the complexity, and for this reason we leave the boundary (see figure region in the bulk enclosed between rays (and possiblytum the state: past and future singularities). Let us startWheeler-DeWitt by patch reviewing is a the regionwith in proposal the respect by maximally to extended Brown blackconsider two et hole the choices AdS-Schwarzschild spacetime al. black defined of hole in inleft time, 4 boundary some dimensions. one as level We on will of denote each the details. boundary. time on the The For simplicity let us first of the WDW patchSchwarzschild-AdS is case, intimately the related spacetime to volumeand the and both action thermodynamic proposals behave volume, should in and work very equally that, similar well. in fashions 3.1 the Review of Brown et al. Wheeler-DeWitt (WdW) patch.late In time, particular, and this we will quantityis see grows a in linearly the contribution first with to halfconsider time this of the at this growth. section possibility that InWDW that the patch the thermodynamical it which volume section is is half the dual of to this the section, complexity. we switch We gear will and show that the spacetime volume Figure 1 When orange). The contributions from the Gibbons-Hawking term are in blue. JHEP03(2017)119 ]), 39 (3.5) (3.6) (3.2) (3.3) (3.4) , it follows from CA- . If we now compare L t R i n E conjecture ⊗ | L ) for the late-time complexification i n ~ A E π 3.4 | ~ it takes for a quantum state to evolve M 2 π / 2 ⊥ n τ ) = E ~ h = E i βE 4 ) π 2 − L R C e ≥ ] and are believed to be the fastest scramblers d dt , t – 12 – ˙ ⊥ L n C ≤ t τ 45 X , ( at late time. That the bound is saturated is another 2 →∞ ψ / lim 44 L | 1 t ( − C Z = i saturated TFD | are the Hamiltonian on the left and right boundaries, respectively, and R H ) is a convincing piece of evidence for CA-duality. This is because it is ]. ]. This latter states that the time 3.4 but is appealing: black holes seem to excel at information-related tasks, since and 46 43 is the average energy of the state. If we take the reciprocal of both sides, and is the bulk action evaluated on the WDW patch. At late L is the thermofield double state: E A H i The Lloyd bound takes inspiration from another bound known as the Margolus-Levitin The complexity of a quantum state is, roughly speaking, the minimal number of quan- TFD rate, we see thatbound, the and that ADM the bound mass is conjecture of the black holethey plays saturate the the Bekenstein rolein bound of nature [ energy [ in the Lloyd which is the Lloyd bound.proved using We point elementary out techniques, that, thethe while Lloyd Lloyd bound the bound with Margolus-Levitin is the theorem a prediction can of be CA-duality ( where re-interpret the left-hand sidecomplexity, (which we then has arrive unit at ofof the frequency) the statement that system: as this the rate rate is of bounded change above by of the the energy review the motivation for the Lloyd bound. theorem [ into a state orthogonal to it is bounded below by: Equation ( reminiscent of aaccording conjectured to upper which the bound rate on of computation the is bounded rate above of by the computation energy. by Let us Lloyd briefly ([ where duality that the rate of growth of the complexity approaches the mass of the black hole: gled, and that the reduced density matrix on eithertum side gates is needed the to usualThe produce thermal statement the state. of state CA-duality from is some that: universally agreed-upon starting point. | The thermofield double state has the properties that it is close to being maximally entan- where JHEP03(2017)119 ], ). 1 38 1 (3.8) (3.7) (3.10) (3.11) ] gives and at 38 B spacetime Our contri- ’ in figure 4 B : ]. . Note that the δt 1 adS 26 V , and S B I 25 ] was questioned by [ B − K | 1 26 1 , V h πG | 8 S x 25 4 p − δt gd ∂M − Z adS √ 0 2 B 1 V πG I Z 8 − 1 πG Λ (3.9) are the null normals to the corner pieces. x 8 2 4 ¯ k ): . Thus, to compute the rate of change of the d 2Λ) + − 2 1 gd 1 d − Σ + − and – 13 – √ R = k ( Kd 1 g V R S Z − Z = (some spatial volume) √ ∝ 2 ]. where V 1 2 πG | M S V 38 8 ¯ k Z S · − − k 1 − | 2 , the WdW patch loses a thin rectangle and gains another thin πG V 1 1 V L denote the upper and lower dark orange slivers from figure S V S 16 S δt 2 = ln − V = remains unchanged. In this section we will follow the more rigorous treatment + 1 a L V A t 3.4 S and to = 1 L V t A δ ). This computation itself, of course, can be found in [ 3.4 : Note that First, since the WdW patch is a region with boundary, the action is the sum of the A technical remark is in order here. The method of computation in [ (all of which are depicted in blue in figure 4 0 where the calculationHowever was the redone conclusion withof a the more boundary term careful as treatment presented of in [ the boundary of the WDW patch. volume Thus, after one evaluates the integralsically above, the we expect product to of see a something spatial which is volume schemat- and the infinitesimal time interval This readily follows from the factHilbert that theory. AdS-Schwarzschild Thus, is a if vacuum webackground, solution evaluate of we the Einstein- Einstein-Hilbert immediately action see on that the AdS-Schwarzschild we have something proportional to the respectively, and that Let us consider firstRicci the scalar difference of between the AdS-Schwarzschild the solution two is rectangles a constant: take into account the contributions localizedThus, at we these see corners (named thatB the Gibbons-Hawking term contributes at the singularity, at rectangle as described in dark orangeaction in we figure have to evaluate thethe action sides above of on these the twoa two rectangles orange detailed are rectangles. null argument except Observe that at thatterm. the all the Also, null singularity, and since boundaries the the do paper boundary not [ is contribute not to smooth the at Gibbons-Hawking the corners of the rectangles, we have to Einstein-Hilbert action and the Gibbons-Hawking(-York) term: When we shift In this subsection, we takeequation a ( closer look atbution the in gravity this calculation of subsectionpressure) CA-duality is arises to to naturally derive from show the that calculation. the thermodynamic volume (together with the 3.2 CA-duality, through the lens of black hole chemistry JHEP03(2017)119 V and (3.14) (3.15) (3.17) (3.18) (3.19) (3.12) (3.13) (3.16) U δt Thus, whatever . Thus, we have B . 3 + r 5 V = β r πr − 4 3 e  = → f , the second term above K + − V V r ) only receives contribution ) (The algebraic details are  and → 3.8 and 3.11 log . In the late time limit, we can U B 2 T Sδt . Thus, in the late time limit, the β 3 B P V δt r L rf e + δt = − r − Mδt 2 → PV , one finds:  + 2 0 3 2 = 3 B U − PV r B GL δt df dr 2 2 adS 2 2 M , Λ = TS Σ = r − − B 3 + and  I tends to r + Λ GL πG = = 2 8 Kd 2 G TS − B 1 B 2 – 14 – 4 S M r − V − A dt Z 3 2 d S explicitly. When we do this, two remarkable things = 2 = = = adS 2 − = 0  2 1 V πG P M 1 V B S 8 V ) that A S I dt 1 S d −  adS − − 1 B V 1 I 1 V S πG 8 S − thanks to boost symmetry of the black hole. adS L 0 , and is therefore invariant under t B I UV coordinate of the 2-sphere sitting at  ]). The contribution of the Gibbons-Hawking term at the singularity is r 38 1 πG 8 is the is a constant. In the late time limit, where B K r Let us now evaluate the remaining contributions in ( Indeed the Schwarzschild-AdS metric in Kruskal coordinates only depends on the coordinates 5 Using the Smarr relation above, the time derivative of the action simplifies to: through their product Next, recall the Smarr relation for AdS-Schwarzschild in 4 dimensions: where vanishes and: Putting everything together, we find the time derivative of the action at late time to be: As for the contribution at the two corners as a measure ofrate the of black growth hole of the interior, complexity. and in thefound same again time, in relates [ essentially it the to ADM the mass: late-time where, in the lastseen equality, how we the used thermodynamic volumePut arises differently, from the the WDW action patch evaluated on provides the an WDW interpretation patch. of hte thermodynamic volume where easily see by inspection ofintegral figure above ( can be interpreted in the language of the extended thermodynamics as: happen. The isalways that cancels the with part the of partthis the of happens the upper for lower rectangle rectangle any quantity which which comes is is out outside outside to the the befrom past future the the horizon, black horizon spatial and hole volume interior. in The equation second ( is that the integral evaluates to: Let now us do the integral for JHEP03(2017)119 is can (3.20) PV V , or equiv- and is the corner P PV ]) is a geodesics TS 34 on the right-hand side is It is straightforward to see 6 M ] where finding the complexity 42 ) (3.21) K PV − poly( TS · + < t – 15 – M ) i 3 2 ) t ( = ψ | ( M by a linear function of time. This is a highly non-trivial C 2 ), in the case of AdS-Schwarzschild, the late-time rate of below ] which establishes a lower bound for the complexity (modulo 3.13 . These observations beg the question of whether 41 V ). To establish that the complexity grows linearly at late time, one needs to also 3.21 On the gravity side, as noted in the introduction already, one can associate various As previously noted, what we are looking for in proposing a holographic dual to the “Late time” in this statement refers to timescales exponential in the entropy. For much later times 6 property of the complexity alonedual. allows for A simple quite illustration some of freedom this in non-uniqueness proposing phenomenon a (given in holographic [ (doubly exponential in theto entropy), zero. quantum recurrence kicks in and the complexity periodically returns Aaronson and Susskind [ the possibility thatdirection an is improbable Nielsen’s statement idea inreduces of to complexity the the theory problem complexity of is geometry finding true). [ geodesics on The ageometrical curved other quantities manifold. to the ERB which all grow linearly in size at late time, so this automatically establishes an upperthe bound thermofield on the double complexity. statebound In the ( particular, usual time-evolving waybound in the quantum complexity mechanics from establishestask, the but there upper are two promising directions. One of them is a recently proved theorem by that the complexity of theof thermofield time: double state is bounded above by a linear function almost by definitionsmallest of number the of quantum complexity. gates needed To build see a state, this, hence recall any way that to build the the complexity state is the complexity is the spacetime volume of the WDW patch. complexity is a linear growth atOn late the time, together information-theoretical with side, consistencygenerally with believed the the to Lloyd linear be bound. true growth but of is the surprisingly hard complexity to at prove. late time is change of the bulkalently action evaluated the on late-time the ratethermodynamic WDW of volume patch gives change the ofserve product the as spacetime the volume basisduality. of for In a the this new, WDW subsection, alternative patch we proposal will is for make the the the case complexity that alongside a with possible CA- holographic dual to the contributions which end upthe on contribution the from horizon the at black late hole time, interior away and3.3 from finally the the singularity. term Complexity with =As volume 2.0 we have learned from ( Smarr relation: as a way tohand keep side track of corresponds thethe to different contribution the contributions from total to the growth, singularity the of the complexity the growth: term WdW with the patch, left- the term with If we now turn the logics around, we can reinterpret the following slight rewriting of the JHEP03(2017)119 : . L K t , and (3.24) (3.25) (3.26) (3.27) (3.22) PV to convert between the CFT  ) . The length of such a geodesic T R R t t | | + R R t L t and t ( + L + ]. 1 2 ] argues that if we think about the CFT t L L t t 34 | 34 | ~ TS K cosh PV TS  ∼ – 16 – ˙ ) = ˙ C ∼ ) C ∼ R R (Spacetime Volume) (3.23) , t , t , we find that indeed the leading term is linear in P L L t . The complexity should again be proportional to the 1 t ~ ( ) = 2 log ( C R ]. To see this, [ C PV → ∞ ): , t 34 C ∼ L L L t coming out of CA-duality calculation as opposed to t seems to be the “correct” energy from the viewpoint of the ( = d qubits, then the complexity grows linearly in time with slope M + M r which should be identified as the complexification rate from the K of the CFT, and use the temperature S TS . While fixed and send PV R t and On the other hand, one could make similar arguments to make the case that the However, we can form 3 quantities with the dimension of energy out of the standard Next, we ask the question of whether “complexity=volume 2.0” satisfies the Lloyd Taking inspiration from CV-duality and CA-duality, we would like to propose now that TS with the entropy , and complexification rate should be number of degrees of freedom, which for a discretized CFT is roughly the central charge To convert between the quantumK circuit picture andtime the and field the theory quantum picture, circuit we time. identify Thus, we find after the translation: viewpoint of quantum circuits [ as a quantum circuit of because we have the mass the Lloyd bound refers to the energy of thethermodynamical system. variables, by multiplying eachM variable by its conjugate.Lloyd Thus bound, we it have: is We will refer to this proposal as “complexity=volume 2.0”. bound. Naively, it might seem that the Lloyd bound favors CA-duality over our proposal, In the late time regime, by design we will have: Another geometrical entity whosespanning size the grows wormhole. linearly As previously atproposal noted, late this by quantity time Brown served et is as the al. the basis known maximal for as an surface CV-duality earlier [ the complexity is dualspacetime to volume multiplied the by spacetime the volume pressure): of the WDW patch (more precisely the is given by (for the case If we keep in the BTZ black hole anchored at boundary times JHEP03(2017)119 M TS TS , (3.32) (3.33) (3.34) (3.28) (3.31) (3.30) , as M M seem relatively PV and TS , (1) numerical factor, while O M  ]  ) (3.29) 2 + 2 2 1 2 47 r 2 + R L t r L 3 − 2 2 3 . + + P 3 + r L L r 3 2 4 L 1 + 1 + ∝ t differ by an PV (   2 ≈ ≈ L G 3 + L + + – 17 – 3 2 4 2 r 2 r r TS PV M TS = = = = = c C and M TS PV M : L and + (i.e. large black holes), we then find r L . Given that there are ambiguities associated with defining the complexity (such have the same order of magnitude. To see this, we express these quantities as become the same quantity ! Thus, for high temperatures at least, it does not make  PV + PV PV We end this section by noting that for most black holes all three quantities r similar connections for chargedwork and of rotating CA-duality, the solutions. situationthe for Unfortunately, gravity charged within calculation and does the rotating not frame- simply black respect holes present the is the Lloyd not computation bound. as offor In clean, the a this and complexity variety section, according of however, to we charged will “complexity=volume black 2.0” holes, and demonstrate that - like in the uncharged case - easy to accommodate. 4 Thermodynamic volume andGiven complexity: the conserved clean charges connectionmodynamic between volume and the the Schwarzschild-AdS complexity, WDW it patch is with natural to the ask ther- whether we can also establish Interestingly, the two quantities and much of a difference whether theor rate as of growth of theas complexity is overall thought of numerical as factors), the discrepancies between For and functions of Thus, one can schematically write down: and the complexification rate at late time is central charge is dualdimensions to we what have we the have Brown-Henneaux formula been [ calling the pressure. For example in 3 bulk It furthermore seems reasonable that the volume would roughly encode the number of sites. times the number of lattice sites. Now by the holographic dictionary we know that the JHEP03(2017)119 (4.3) (4.1) (4.2) dt R t + 2 dimensions. The metric ! 1 n 2 n 2 2 − q r Ω L n d r 2 + r − , the patch loses a sliver and gains another 1) + 1 L + + - - − r r r r 2 − ) q n δt 2 q n + r 2( r ( + dr r f L

t + + 1) 1 2 – 18 – 1 − − − dt n n n , we depict their Penrose diagram together with + + - - ) r r r r n r ω r 2 ( 2( f − − r is shifted to L = = t 2 L A )) = 1 r L ds t ( +δt L f t . The Penrose diagram of a charged and/or rotating black hole and a Wheeler-DeWitt On the gravity side, for both charged and rotating black holes, the Penrose diagram 4.1 Electrically chargedLet black us holes starttogether with with the the Reissner-Nordstr¨omblack gauge hole field are in given by: is qualitatively thethe same. WDW patch. InSchwarzschild-AdS Note figure solution: that the the upper part WDWbut of approaches patch the the is patch inner qualitatively no horizon different longer at runs from late into time. that a of singularity, the one (depicted in darker orange). The singularity is inthe red, thermodynamic and the volume horizons and aregrowth. the dashed. We pressure will emerge relegate naturallythe the from next interesting the section. question late-time of rate consistency of with the Lloyd bound to Figure 2 patch (depicted in orange). When JHEP03(2017)119 (4.7) (4.9) (4.4) (4.5) (4.6) ). (4.10) 4.6 ... ] for the outer horizon 8 2 ) +  R t dt 2 + r J L 2 t between the two thermodynamic )( − ) n − 2 − r (Spacetime volume) (4.8) dφ V L P +1 − 2  n  − 2 r ~ n Q ∝ + r ) r 2  + 4 π g ( n V + ) ( S difference πr − − n ) + 1 2 P √ S r = 2  n ( n x dr – 19 – Vol( = f n  πr d ˙ V Vol( C = + = Z 2   2 V →∞ dt 1 V lim ) L L t r ( ∝ f − = Hilbert 2 − of course refers to either horizon. The spacetime volume of the ds  Spacetime volume = Einstein S , and the late-time rate of change of the complexity still takes the form ( − V − + The thermodynamic volume can bevolume): found to be (see for example [ Thus, like for thewithout Schwarzschild-AdS the case, boundary the term)simple distinction case and between of spacetime the the volume rotating bulk BTZ is action black not hole, (i.e. very the metric important reads: here. In the Next, we move onholes to are vacuum discuss solutions rotating ofon-shell holes. the Einstein-Hilbert Einstein-Hilbert action Like action, (ignoringpressure the and boundary multiplied this Schwarzschild-AdS by contributions) again case, the is implies rotating spacetime that proportional volume the to of the the WDW patch: But in the end,V the second term on the right-hand side4.2 above drops out Rotating of black the hole difference Note the slight differenceplexification compared with rate the is Schwarzschild-AdS nowvolumes. case: proportional Let the to us late-time the end com- BTZ this black subsection by hole. mentioningsomewhat the different Here 3-dimensional form: there case of is the potential charged for some surprise, since the volume takes the recognize the difference betweentime, thermodynamic then, volumes we in have as the advertised: equation above. At late where the ellipsis stand forthe terms which time are time-independent derivative (and of therefore the drop out complexity) from or are exponentially suppressed at late time. We the geometric volume of a ball in flat space: where the subscript WDW patch takes the form: As mentioned in the introduction, the thermodynamic volume is well-known and looks like JHEP03(2017)119 (4.18) (4.19) (4.20) (4.17) (4.12) (4.13) (4.14) (4.15) (4.16) 2 θ dθ ∆ 2 ρ +
2 − r r dr 2 ∆ 2 a ρ − + 3 − 2 r  ]: dφ − 2  2 θ 48 2 2 ) + 2 a 2 ) (4.11) /L r 2 a 2 2 − /L + + g a 2 a θdφ V 2 + ) sin a A 2 2 + a 2 − + − r + 2 + a − 1 r r − sin 3 + + 1 1 3 r +  a 1 V ( 2  mr + – 20 – ) = − r 2 P π 2 ( πr + a dt − = V 4 3 + θ ) ˙ − = 4 C 2 2 1 2 = r (∆ + dt 2 L 2 + A θ ( g ) θ 2 V G θ 2 2 g )∆ 2 2 2 g 2 2 a r − cos a 2 )(1 + mr cos 2 7 − 2 g 2 = g 2 − a 2 a (1 1 L a A + 2 + d ρ dt (1 + 2 − 2 r ]), and there are two possible notions of volume one can identify. For r − + 36 = ( = 1 = = is the ratio of the angular momentum to the mass. The late-time growth r θ 2 2 ρ ∆ ∆ ], we refer to the volume in the non-rotating frame as the thermodynamic ds 36 J/M ] but not the first law. On the other hand, the thermodynamic quantities derived in the non- is the area of the horizon: 36 = A a We also note here that the thermodynamic quantities derived in the rotating frame obey the Smarr 7 relation [ rotating frame at infinity doIyer-Wald formalism. obey a first law (in addition to a Smarr relation) and can be derived from the where Putting the two equations above together, we have: Following [ volume and the onegeometrical in interpretation: the rotating frame as the geometric volume. The latter admits a which again is proportional touum the solution. spacetime As volume for bydepending the virtue thermodynamic on of the volume, whether we solution the have being analysis two a is different vac- done notions of in volume a non-rotating or rotating frame at infinity. Here of the bulk Einstein-Hilbert action was computed in [ thermodynamics is somewhat differentodd depending or on even whether (see thesimplicitly, [ spacetime we will dimension focus is on the 4-dimensional case. The solution is given by: tional to the difference between the two thermodynamic volumes: Next, we move onKerr-AdS). to This discuss case the is case substantially richer of and rotating more black interesting, hole as in the higher analysis of dimensions the (the After some calculation, the late time rate of complexification is again found to be propor- JHEP03(2017)119 = 3, (4.21) versus ,L a = 2 - -V M + V and for each value 60 a = 1, blue is for 50 ,L associated to the inner horizon − = 5  V 2 M 2 a ) (4.22) /L 40 In this section we seek to answer this 2 − + a V 8 2 − − = 0, which reduces to the Schwarzschild case, − r 1 a +  V ( − – 21 – 30 P ]. πr = 4 3 ), and converting from the bulk action to the space- 39 ˙ = 1, but this is modified by the AdS length dependence of C a = : 4.20 , we plot the angular momentum-to-mass ratio − + . Notice that V V 3 20 − V in − . − ) and ( r L + , we vary the angular momentum to mass ratio V L → 4.17 = = 2. and 10 + and r V ,L M . As we approach extremality, which here occurs as the plots flatten out on the M V = 1 tends towards zero. In the plot green is for M V a 0.2 0.6 0.4 1.2 1.0 0.8 for fixed . Given − V We thank Adam Brown for pointing this out to us. − 8 + fashion, and one is naturallythe led other. to One ask whether advantagetheories is any advantage that could can there have be are problems identifiedquestion no for near as boundary one a regards terms, or the singularity. which Lloyd in bound higher [ curvature 5 Action or volume? In this paper wecomplexity of have discussed the boundary twoon different thermofield the possible double one state. holographic handwith identifications On with the of could spacetime the the identify volume action this of of complexity the the same. Wheeler-DeWitt These patch, two and quantities on behave in the a other rather hand similar time volume, we finally find: To help gain intuition, inV figure by the replacement Comparing equations ( has the maximal left (In flat space exremalitythe occurs metric), for and red is for As in the previous cases, we can define a second volume Figure 3 solve numerically for JHEP03(2017)119 (5.4) (5.5) (5.1) (5.2) on the R µQ are the radius − e R M H R i i n and µ e Q r ! − ) 2 n TFD | i E R | t gs δM ) L ) ( i R n O µQ on the left, and µQ Q + − n L − ) (5.3) R E | . This is because the electrostatic potential is H ( 2 M µQ δM R µQ i / ( ) − H L + n − e − 1 L L 1. Expanding the outer horizon radius near evaluated in the appropriate ground state, ) t µQ 1 it corresponds to some extremal black hole ) − M and H + 3 ( L 2 n L µQ √ µ ~ – 22 – E µQ µ > 2 H µQ ( π µ > − + β p − 2 L − e µ M H ˙ ( ( i C ≤ M h n − between X e ~ 1 + 2 π µ Z

= 1 1, but for e i √ r ) ˙ ≤ C ≤ R is the total mass of the black hole, and = ≈ µ , t i + µ L M r t , ( e ψ : | is nothing but M µ TFD = 1). | − gs ], the existence of conserved charges puts constraints on the system and ) G 26 M µQ := − M δM 9 Note the difference in the sign of 9 = 1. First we consider the case where Where and mass respectively of an extremal black holepositive with on the one same side and chemical negative potential on as the other. the hand. For simplicityG we restrict our attentionextremality, we to find 4-dimension, that and work in units where AdS for black holes with (In units where 5.2 Bound violation:Now near we the will ground state check to see whether the Lloyd bound is obeyed by the two proposals at where ( which will be eitherconsideration. empty If AdS we think or ofnatural our an system to extremal as take being black the in hole theis ground grand depending the state canonical same ensemble, on to it as correspond the is the most to case black the under hole geometry under whose consideration. chemical As potential it happens, this is nothing but pure This however violates our intuitionrate that of as an zero extremal temperaturethus black system, seems hole the appropriate should complexification to be modify zero, the above and to that the bound should reflect this. It Based on this, one would guess that the appropriate generalization of the Lloyd bound is: This state time-evolves by theright: Hamiltonian In this subsection, letAs us argued derive in the [ implies Lloyd that bound the in rate thewithout of presence charges. growth of of Let a thechemical us conserved potential complexity start charge. at by late generalizing time the is thermofield slower than double in state the to case include a 5.1 The Lloyd bound with conserved charge JHEP03(2017)119 and (5.6) (5.7) (5.10) (5.13) A ˙ C # ) ) (5.9) 2 3 ! ) 2 δM M δM ( (  O δM O 2 2 ( ... + µ µ + O + ) (5.11) 2 − 5 3 + 1 1 + δM ) (5.8) ) (5.12) M 3 2 M δM 3 (  2 L 1 , and so the lower order behavior δM 1 O M L δM ( 1) L − 2 M = 2 ( − + L 2 2 O − 2 3 1) O 4 µ µ µ M 2 µ + 8 + 1) − M + 2 µ p  p 2 2 2 3 M 3 (2 L µ µ
√ 2 ( 2 δM 3 √ 6  3 + 1 µ 9
2 2 µ 2 2 ≈ ≈ 1 (2 √ πµ 1 – 23 – 1 µ µ µ 4 e − ) + 1) − ) − − − " 1 2 − 2 2 2 3 e 1 1 + µQ µ µ µ r 1 µQ µ 3 (  2 −  p π ≈ − 2 2 2
≈ µ M ≈ ≈ ≈ M 3 − ) ( r under both proposals as − 2 ( A V ˙ ˙ − ˙ 1 C C µQ − µQ − C ) 1, in which case we expand around empty AdS. Here the bound

3 + − A − e r ˙ r ≤ C 2 µQ − M r µ Q ]. P ≈ − 2 ( 26 − = = r M ( A V ˙ ˙ C C becomes under each proposal ˙ C Next we consider exactly saturates the bound to lowest order in A ˙ As this term isAdS. positive definite, This we would seethough that seem the the to bound expectation put is that CV-dualitythis violated in case. the as a bound we slightly approach should empty better be position saturated, than or CA-duality, nearly so is not met in C becomes important. Expandingthe the bound) bound directly violation we (i.e. find to the lowest difference order between We see immediately that thebut bound the is bound satisfied is in far CV-duality sufficiently from near saturated.The extremality, situation with CA-duality is a bit more complex: But in agreement with [ becomes to lowest order On the other hand the bound is given by We thus see that both proposals must violate the Lloyd bound near extremallity. This is From these we can expand one we are considering. We likewise may expand the inner horizon as JHEP03(2017)119 ]. 48 (5.18) (5.19) (5.20) (5.21) (5.22) (5.14) (5.15) (5.16) 0 ≤
)  4 + ) − 4 − r − r r r 5

5 − + 2 4 − − 2 + 3 + − r r 2 + 2 + r + r r + r − + 2 − 2 − ( r − r 1. r ( 2 + 2 − − 2 2 ) (5.17) + r − r r ) 4) 2 ≤ r r − 2 − + 2 − − L r − + ) r + r r µ − r 4 r 2 4 + d 2 − − π r − r ) − L − ) 3 ) + − + + 2 2 + + ) + r r r 2 − 2 + 4 + r 2 − ( + − r ( r r + (3  2 2 2 − + r r d − 2 πr − 4 − L 2 + L L + 2 − + r 3 − 2 + r 3 r − 8 2 r 2 L r r d
+ 2 − + L 2 ≤ 3, we find that ( r L + 2) − r d r + 2 − ≥ L d L ) r − 8 r r 16 + − + 2 2 ) ( 2 ( r − 2 − 2 2 r + quantity, we find in 5 dimensions r 2 2 + 2 r − 3 + r − r d > L π L 2)Ω r r + + + 2 + + + πL 2 − ⇒ r r 2 − + 2 − CV 8 r 2 − r + 2 r 1 ) + + (3 d + r ( ) = L 2 3 ( 2 4 + + − + ) 2 – 24 – − L L ≤ r r r r ( 2 − 2 µQ 2 2
r 2 − − L + ) = L 2 + ( − π 0 − πr − − r 2 + 3 3 − 4) r 3 r r µQ + ( ) ≤ M ( +  − 2 r − respects the bound whenever − 1. The exact expressions in 4 dimensions are − 2 ( − s 2( 3 2 − − r L r π 2 + r r L 2 V − 2 ≤ r − ) = = 2 − M (3 2 C L + + ⇒ µ  = A µ 2 + L 2( 2 ˙ 1 3 + µQ r ) = 4 π π C ( r L 1. Now the exact expression for the chemical potential is 3 3 − ≤ − − − µQ A − ≤ 2 2 − ˙ C M µ ≤ = − µ ≤ − = πr ) 2( ) 3 M − µQ = µQ 2( space where V 2 − ˙ − − µ C − V M , r ˙ M 1 case, one can in fact do better. We can find the bound violations as an exact C + 2( 2( r ≤ − − µ V V ˙ ˙ Generalizing further to arbitrary dimension C C and from which we get Which is a positive definiteBeing, quantity. for This now, in a fact bit recovers a less result ambitions already with derived the by [ and so we may conclude that and so The second expression is clearlyis positive always violated definite, for and so the under CA-duality the bound From this we may conclude that and In the function of the innerregion and in outer horizon. Note of course that these are only valid over the 5.3 Bound violation: exact results JHEP03(2017)119 (5.23) (5.24) (5.25) ]. Even though 1 for all AdS-RN 1. We conjecture 50 , ≤ < 49 2 ) 2 µ − µ r − leaves all the matter fields + g r ( 2 0 (5.26) L < 2 − 2 L ) 3 − 2 L r − 2 r . . + = 1 for the AdS-RN case we find: ~ ~ ≤ − + E α π π r 2 ) αE ( − ≤ ≤ µQ r – 25 – ˙ ˙ + C C − r M ( ) = − µQ V ˙ ) again, we find: C − . For example, for M 5.17 ( α − V ˙ C = 1 is also consistent with the bound. We leave further explorations of to future research. α α It would seem, however, due to the arguments leading to this bound, that the bound unchanged. In comparison, we dogeneric not feature have of this these luxury spacetimesexplicitly in is on the the the case fact cosmological of thatthese constant Lifschitz the solutions: (this spacetimes profiles property a are of is not theaffect somehow matter asymptotically the related fields AdS), matter to depend fields. so the fact varying that the cosmological constant will BTZ black hole andseveral the interesting and R-charged intriguing blackthat features hole. it of the will In thermodynamicsolutions the prove in volume, latter useful the and future. case, to we Of ourthe compute particular believe interest computation method the are of Lifschitz explains volume black the holesrelatively of volume [ simple for many since the more we R-charged complicated saw black that hole black was perturbing hole a the bit coupling involved, it is still Let us summarize theIn main the findings first in part thisviewpoint of paper the of and paper, the sketch we Iyer-Waldpresent out analyzed formalism. a some the systematic notion future Using way of directions. to thermodynamic a derive volume slight the from generalization volume the and this illustrate formalism, it we in two cases: the charged Hence the value other values of 6 Conclusion and, using the inequality ( under which proposals and sets of circumstances does hold for various values of should only be trustedhow up robust to the an above overall discussion factor. is under It the would insertion be of interesting, some therefore, pre-factor. to see For example, without proof that CV-dualityspacetimes. repects the Lloyd bound whenever 5.4 Altering theWe bound have by considered a the Lloyd pre-factor bound in it’s usual form, And so CV duality in 5 dimensions respects the bound whenever JHEP03(2017)119 ]. Tensor Phys. Rev. 32 , ]. SPIRE IN [ ] with an eye on the 54 – 51 (1973) 2333 D 7 Phys. Rev. , – 26 – ], the black hole interior is an example of emergent 56 , 55 ), which permits any use, distribution and reproduction in ]. SPIRE IN CC-BY 4.0 Black holes and entropy Generalized second law of thermodynamics in black hole physics [ . Thus, one can hope that the pressure and volume variables can prove This article is distributed under the terms of the Creative Commons (1974) 3292 par excellence D 9 How to take this story further? As mentioned in the introduction, a tensor network In the second part of the paper, we related the thermodynamic volume to the holo- J.D. Bekenstein, J.D. Bekenstein, [1] [2] References PHY-1620610. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. We would like touseful thank comments which Adam they provided Brown, onto Bartlomiej an thank early Czech, draft Kimberly of and Carmona this Leonardis paper. for based Susskind assisting We would upon in for further proof-reading work like the supported the manuscript. by the This National material Science Foundation under Grant Number to black hole complementarityspace [ helpful to our understanding of quantum gravity in the future. Acknowledgments picture of the black hole interiornetwork was introduced is by Hartman a and Maldacena topicemergence in of of [ spacetime. much Thus recent onebe can interest understood ask for the in question: holographers the can language [ the of pressure-volume variables tensor networks or quantum circuits ? Also, according thermodynamic fashion aspressure. the partial The volume derivative computed ofquantity. by the We the conjecture ADM Iyer-Wald that formalism masstime this is with derivative should by of further respect construction the correspond the toseveral spacetime to same examples, the volume the but of late-time have the value no Wheeler-DeWitt of proof patch, the that and this have holds checked generally. namic volume (if there(if is there only are one twopatch horizon) horizons). (ignoring or On boundary the the difference contributions)volume other of is and hand, thermodynamic the charge-potential. the volumes sum Thevolume bulk of presented several action in “work different evaluated this terms” ways paper on to involvingthe may the be pressure- arrive relationship WDW a at between little the them: confusing thermodynamic to the the reader, thermodynamic so volume let may us state be again defined in the usual graphic proposals for thevolume complexity. (together with In its particular, conjugateof the we an pressure) showed is eternal that intimately AdS the relatedintimate to black relationship thermodynamic the hole, can WDW be and patch of stated this change of cleanly holds the in for WDW two a spacetime different volume large ways: in class the on of late one time AdS hand, limit black is the holes. precisely rate the This thermody- JHEP03(2017)119 , ]. , ]. ] Class. D 26 , SPIRE , SPIRE IN , [ IN ][ (1975) 199 Gravitation from ]. ]. (2015) 073 43 ]. 12 (2015) 999 ]. (1976) 191 SPIRE SPIRE arXiv:1507.06069 93 IN [ (2014) 205002 (1995) 4430 IN (2015) 221601 Int. J. Mod. Phys. SPIRE ][ ][ , JHEP IN 31 SPIRE D 13 ]. , ][ 114 ]. IN D 52 arXiv:1312.7856 ][ [ ]. SPIRE (2015) 184 SPIRE IN IN Can. J. Phys. 09 ][ Phys. Rev. , ][ SPIRE Commun. Math. Phys. , IN gr-qc/9403028 Phys. Rev. , [ (2014) 051 , ][ Lower-Dimensional Black Hole Chemistry arXiv:1409.3521 Locality of Gravitational Systems from ]. JHEP [ Phys. Rev. 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