JHEP09(2016)154 Springer August 4, 2016 : September 18, 2016 September 27, 2016 : : . We offer a general 2 Received 10.1007/JHEP09(2016)154 ), with a metric that can 2 Accepted Published doi: a , Published for SISSA by and Claire Zukowski a,b [email protected] , . 3 Nele Callebaut a 1604.02687 The Authors. c AdS-CFT Correspondence, Conformal Field Theory, Spacetime Singularities

Recently, two groups have made distinct proposals for a de Sitter space that is , [email protected] Instituut voor Theoretische Fysica, K.U. Leuven, Celestijnenlaan 200D B-3001E-mail: Leuven, Belgium [email protected] Department of Physics, Columbia538 University, West 120th Street, New York, NY 10027, U.S.A. b a Open Access Article funded by SCOAP ArXiv ePrint: infinity. For the BTZ blackthat string, both corresponding prescriptions to lead amethod to thermal for state an relating of emergent kinematic the hyperbolicvacuum space CFT, and patch and we thermal of the show cases. dS auxiliary de Sitter space thatKeywords: is valid in the turbations around the vacuumthese state two of emergent spacetimes CFTs. in the Wekinematic vacuum provide spaces case of support and nontrivial beyond. for solutions theblack In of particular, 3d equivalence hole, we gravity, of including and study the the conicalglobally BTZ hyperbolic black singularities. string, spacetimes BTZ that We support argue dynamics that given boundary the conditions resulting at future spaces are generically emergent from conformal field theory (CFT).holographic The CFTs, first proposal the is kinematic that,ically for space two-dimensional anti-de of Sitter geodesics bulk onbe is a derived two-dimensional space-like from de slice the Sitter of entanglement spacede entropy the of (dS Sitter asymptot- intervals dynamics in the emerges CFT. naturally In from the the second first proposal, law of entanglement entropy for per- Abstract: Curtis T. Asplund, Equivalence of emergent deconformal Sitter field spaces theory from JHEP09(2016)154 5 ]. 11 – 7 , 1 ], boundary entanglement 6 33 – 4 – 1 – 27 23 34 24 27 28 ]. The growing consensus is that at least outside “shadow” 23 14 3 8 – 1 9 7 26 20 3 3 1 30 4.3 Thermal state 5.1 Refined prescriptions 5.2 Beyond universality 4.1 Vacuum on4.2 a plane Vacuum on a cylinder 2.1 Global AdS 2.2 Poincar´epatch 2.3 BTZ black2.4 string BTZ black2.5 hole Conical singularity entanglement” program, a keymula, tool which for relates addressing entanglement this entropybulk has in extremal been the surfaces CFT the to [ Ryu-Takayanagiregions for- the blocked areas from of Ryu-Takayanagientropy surfaces boundary-anchored reconstructs by the barriers bulk [ spacetime and Einstein’s equations [ The AdS/CFT correspondence provides atum) powerful equivalence gravity between a in theory asymptotically(CFT) of in (quan- anti-de one Sitter lower dimension. spaceity In over (AdS) the spite last and of two great a decades,the progress the conformal field in fundamental theory question our field has of understanding not how theory of been bulk the fully geometry dual- answered. emerges from Within the nascent “emergent spacetime from 1 Introduction A 3d gravity B De Sitter embeddings and penroseC transformations Two-sided BTZ black hole 5 Discussion and further directions 3 Causal structure 4 Relation to auxiliary de Sitter Contents 1 Introduction 2 Kinematic space JHEP09(2016)154 ] ], a 21 , 20 ]. One 16 13 , 12 , the geometry of 3 – 2 – )). ) and extend the analysis to include the effect of ) of a constant time slice of AdS 2.1 ], which matches earlier results related to differential C 2 ] proposed that the MERA tensor network is actually H 16 , -dimensional de Sitter space. Unlike the kinematic space 16 d , 15 15 . The geodesics in these spacetimes can be mapped by large 13 , rather than of the bulk slice itself. Importantly, unlike the two- 3 and -dimensional CFT in the vacuum with a ball-shaped entangling region 5 d We apply the kinematic space prescription to additional nontrivial solu- ], calculates the kinematic space metric solely from boundary entanglement 19 – . The prescription in [ 17 2 )). By applying the entanglement first law for small perturbations around the ], and the authors of [ 4.1 14 As a maximally symmetric solution to Einstein’s equation in the vacuum, it is perhaps In a second construction that we refer to as the “auxiliary dS prescription” [ The first approach is rooted in an attempt to obtain the discretized geometry of a space- One unforeseen consequence of this program was the identification of an auxiliary the conical singularities, we showhyperbolic that patch the and resulting kinematic glueddepicted spaces together are, in “sub-de respectively, figures the Sitter spaces”diffeomorphisms to of parent de geodesics Sitter inall space, AdS, subregions and which of for are the this original reason their de kinematic Sitter spaces space. are What is less obvious is that these kinematic geometry. In thegraphic latter CFTs, two while cases, in thequotiented all results BTZ other are black cases non-universal hole, wefor and consider we the apply they find two-sided only are case good tophase universal. (see agreement holo- transitions In appendix with in the the the casematic partial space. entanglement of results entropy, In the the which of other we [ cases, show the results result are in entirely new. defects For in the kine- BTZ black string and kinematic space and auxiliary dSof pure prescriptions AdS. agree for bulk 3d gravity solutionsOur outside results. tions of 3d gravity: the (1-sided) BTZ black string, its quotient, and the conical singularity proposal, this construction isand intrinsically as dynamical. the authors It stress, applies it is in independent arbitrary ofnot dimensions, the surprising standard to AdS/CFT see correspondence. need de not Sitter appear be in related different beyond arenas. the A vacuum priori, case. the two It constructions is thus a nontrivial check to see if the de Sitter spaceHamiltonian propagator of is a recognized(eq. hidden ( within thevacuum, expression the for authors the demonstrateGordon modular equation that in an the auxiliary entanglement perturbations satisfy a Klein- this emergent kinematic space isspace Lorentzian. dS Specifically, it isentropy a [ two-dimensional de Sitter entropy of intervals in the CFT (eq. ( known as the Multi-Scaleschool Entanglement of Renormalization thought Ansatz has (MERA)e.g., pointed [ [ out severala challenges discretization of to a the “kinematicspace-like consistency space” slice of of of boundary-anchored AdS/MERA, geodesics AdS dimensional contained within hyperbolic a geometry ( Lorentzian geometry from CFT entanglement data,In distinct from fact, the usual there bulk have AdSfrom space. recently CFT. been two distinct proposals for anlike slice emergent of AdS de from Sitter a boundary space tensor network (ansatz for the ground state wavefunction) JHEP09(2016)154 ] . , , 4 B 23 , 2.3 (1.1) 22 and A .) In section /π. C S . It also suggests a explicitly reinstated , we summarize our = 5 L L 5.1 interval is equal to the 2 with spacetimes in their own right. , we first review the kinematic 2 . In section , which we fix to be the circumference dudv 2.3 ) L u, v ( ). (For a review of the 3d gravity solutions ], with a length scale . 3 ent ∂u∂v π ], can be defined for any CFT in any state. S u, v – 3 – 2 ∂ 16 2 , causally well-behaved L 15 c 12 = ) of a CFT interval [ u, v ( of CFT on cylinder )). This patch is precisely the kinematic space we obtain in section ent 2 K bulk spacetime, where the kinematic space has a geometric interpretation S 3 ds 4.21 The paper is organized as follows. In section (eq. ( ], the modular Hamiltonian associated with a thermal CFT 2 In the final stages of preparing this paper, we learned of other upcoming results [ For the BTZ black string example, we construct an explicit nontrivial match between The global hyperbolicity of these kinematic subregions suggests that the space of 20 of the cylinder’s compact dimension over 2 Kinematic space Kinematic space, asHowever, formulated our main in interest [ istotically in two-dimensional AdS holographic CFTs whichas have a an asymp- space of boundary-anchored, oriented geodesics. proposed refinement of the twoand constructions. areas We conclude for with future a work. discussion of our results that overlap with our work. structure of the resulting spacetimesand the (section embedding coordinates and Penroseand transformations for used, see the appendices relationwe to review existing the work auxiliary on destate, the Sitter providing BTZ construction a and quotient match extend see with the appendix the construction to results the of thermal section The length scale corresponds to theS de Sitter radius Outline. space prescription in the casespace of of pure the AdS. BTZ We black use string, the black prescription hole to and derive the the conical kinematic singularity, and discuss the causal ment entropy on the right hand side: of [ integral of the energyof density dS times aThe Klein-Gordon matching propagator informs on our the formulationrefinement of hyperbolic of the the patch equivalence kinematic in space section prescription for a CFT on a cylinder from the entangle- iary de Sitter proposal,tion. which Furthermore, is the defined factto intrinsically that in future the kinematic terms infinity spaces is ofbulk always consistent dynamics propagator. have with and a the propaga- Cauchy surface boundary close conditions required forthe a two boundary-to- emergent spacetimes. We demonstrate that in a direct extension of the result Indeed, we present arguments that theyconditions are globally can hyperbolic be spacetimes set whose boundary at future infinity. geodesics can bepropagate. interpreted as It a provides background an immediate spacetime consistency on check which for dynamical the fields matching can with the auxil- spaces are not just subregions, but are JHEP09(2016)154 , . ) ], 1 19 u, v (2.2) (2.3) (2.1) u, v , rep- ( v > u 1 and ent u, v, θ, α S 18 = 0. This t ], with v, u will define time-like θ shown in figure . By [ ) coordinates of figure α v spacetime, there is a time and θ, α 3 kinematic space prescription α and , depicted in figures θ 2 H = 0) is proportional to the length of the geodesic. These are related ) and ( t . A constant time slice of a locally α . The geodesics or equivalently their t ). The null coordinates in kinematic = 0 and includes, in particular, static v u, v t 1 dudv . ) ] proposes a and α , α . ]. Note that we stick to a single orientation 15 u, v u − + ( θ θ u, v ent – 4 – ∂u∂v ). S = = 2 1.1 v ∂ u ] in the CFT (at = 2 u, v We will use the ( ds and the opening angle θ ], signifies the interval going counterclockwise from the point to the point with angular coordinate π 1 2 u , [0 ∈ ]. We take kinematic space to refer to the set of these geodesics for all 24 u, v , 3 , such that all space-like extremal curves that anchor on boundary points with 1 and t v will be lightlike kinematic space coordinates, while ≤ v bulk will have a 2-dimensional hyperbolic geometry u The Ryu-Takayanagi holographic entanglement entropy proposal states that the en- By invoking results in integral geometry, [ Given any time-reflection symmetric asymptotically AdS 3 We will in practice use the form given in ( 1 and = 0 are entirely confined to a space-like slice defined by the condition coordinates are not angularu but have dimensions ofand length. space-like coordinates, Throughout, respectively. thestate Their coordinates under detailed consideration. specification depends on the CFT with angular coordinate we mean the closure offor the intervals complement on of the [ circle,terclockwise while orientations. we will We consider of willPoincar´epatch AdS geodesics have and with non-compact the both BTZ directions black clockwise in string, and in the which coun- cases discussions the of corresponding the A comment about notation. to label a CFTnot. interval, In both the whenwith case it of is a defined compact, on circular a direction, compact we direction will and use when the it convention is that [ resenting the midpoint angle to the endpoint coordinates by tainment relation of boundarytime-like intervals: separated, otherwise two they geodesics arethey contained space-like share within separated a or, one in left anotherspace the or are are marginal right the case endpoint, where boundary null endpointboundary (see coordinates intervals figure can also be specified by the coordinates for deriving a metric onof kinematic the space boundary entirely intervals: from the entanglement entropy This spacetime is Lorentzian due to a natural causal structure inherited from the con- tanglement entropy of an intervalof [ the (unique) boundary-anchoredthe interval geodesic [ with minimalintervals of length the that CFT. is homologous to coordinate t follows directly from the reflection symmetryspacetimes, about which would haveAdS this property at each JHEP09(2016)154 (2.7) (2.4) (2.5) (2.6) . The 3 and opening θ ] 25 , in conformally . L ) ), it takes the form π ) is [ 2 u 2.3 ≤ − . )–( θ v ( ) ≤ π 2.2 R 2 0 with radius = ] ≤ 2 π, L 26 , ≤ u, v Σ , or by its midpoint angle πL α v ≤ ≤ . sin (0 and 3 through [ ` Σ π u G 3 3 2 G
– 5 – log for a review of the conformal boundary). The ) (0 = u 2 c 3 2 − c v A dθ = 2 dudv + ent 2 sin S 2 dα is the central charge of the CFT, related to the AdS radius L ) is universal, i.e., it only depends on the particular CFT ), we find − c ( = 2.4 1.1 α 2 2 2 (see appendix L ds sin R π = 3 2 ds . (a) A boundary-anchored geodesic on a constant time slice of AdS, with boundary is the UV cutoff, and . (b) These geodesics enjoy a natural causal structure based on the containment relations  α From the prescription ( and 3-dimensional gravitational constant This is the metriccompactified of null coordinates. a In 2-dimensional the de coordinates Sitter defined space in ( dS We note that thethrough formula the ( central charge. where ` We begin bydual reviewing is the the results vacuumcircumference of state Σ the of = prescription a 2 boundary in CFT entanglement entropy on the of a case an cylinder interval of of with pure length compact AdS space-like dimension of separated if they have embeddedright boundary endpoint intervals (top (top left), right), null(bottom and separated two). space-like if they separated share if a their left boundary or intervals are not2.1 embedded Global AdS Figure 1 interval in thick purple.parametrized by the The coordinates intervalangle of or its endpoints, equivalently theof geodesic their anchored boundary to intervals its (colored endpoints in is thick purple and thicker green): geodesics are time-like JHEP09(2016)154 ) ) b. at 2 3 2 θ, α (2.8) (2.9) geodesic 2 in dS and points on de Sitter H p p 2 are located at . H 2
π H hyperboloid (blue). 2 2 ) shown in figure , and can be identified H ≤ 2 θ B.1 ≤ 0 , ∞ by associating a point 2 ] is mapped to a point ). The two orientations correspond α . < t < u, v ], which in a pure state share the same − 2 α v, u −∞ on a CFT π, π 2 + θ log tan – 6 – ] and [ )( − 2 ) of kinematic space, which is the topmost line in u, v = corresponds to a geodesic below the waist with the of the geodesic is a natural time coordinate on the a. The mapping between geodesics on t dθ t 3 2 π 2 α → ∞ = t ( α + + cosh a, a boundary interval [ I 2 2 ], or equivalently, to the corresponding boundary-anchored dt − ), this covers the full de Sitter hyperboloid ( u, v are intersections of origin-centered planes with the 2 2 B.3 L H = 2 in de Sitter at the tip of the lightcone extending into the bulk. (b) In embedding ds p . Kinematic space defines an emergent dS is also intuitive from the embedding diagram, since geodesics on 2 As presented in figure To observe which portion of de Sitter is covered by these coordinates, we convert to ]. (a) The CFT interval lies at the asymptotic future boundary of dS u, v in dS its intersection with aspace plane via centered the at two the normaland origin. one vectors on These of the map the contracting to plane:but region two have of one points opposite de on Sitter. in orientation. the de Such expanding geodesics Sitter region share (see the same figure radial profile Ryu-Takayanagi curve and boundary entanglement entropy. the tip of thecorrespond lightcone to that points projects onwith to the the the asymptotic interval. boundary, future hencethe Geodesics Penrose the with diagram conformal shown zero in opening boundary figure angle can be identified By the embedding ( The expanding and contracting portionsin are the equivalent up region to above orientation:same the a bulk waist geodesic profile but ( oppositeto orientation, complementary ( boundary intervals [ The metric becomes In other words, thespace opening of angle geodesics. global coordinates by the transformation [ with a point space, geodesics on Each such plane specifies a point in the dS hyperboloid via its outward pointing normal. Figure 2 to a given CFT interval [ JHEP09(2016)154 2 is by c + I . The (2.10) (2.12) oriented hyperboloid length 2 geodesics with 2 rather than H ). . − in planar coordi- I ) 2.5 , represented in (c) as L 3 ∞ lines (dashed). The dS ), the kinematic space θ 2.4 ) (2.11) < θ < ∞ ), and (b) as a dS , B.9 −∞ u of the interval, ,  − 2 . α ∞ v < u, v < 4 ) and ( log B.8 α < −∞ c 3 ( ≤ , the single-interval boundary entanglement = L  L is a 2-dimensional de Sitter space, represented (a) as – 7 – lines (black) and constant 2 3 )  and “radius” t u log θ ) (0 c 3 2 − du dv v dθ = ( 2 + L ent 2 S dα = 4 − 2 ). ( , with constant 2 2 2 , ds 18 1 α L R = 2 ds are again the interval endpoints, but now with a dimension of 2 is highlighted in thick black to stress that kinematic space is the space of v ) will reduce to π/ ), which cover the planar patch, see figure = 2.4 and . The kinematic space for pure AdS α B.4 u Because of the universality of the entropy formula in eq. ( In all the cases we consider, we could equally well cover the patch connected to 2 This is thenates metric ( of a 2-dimensional de Sitterflipping space the with sign of radius tation. the time coordinate. This would correspond to giving the geodesics an opposite orien- or, in the coordinates of the midpoint where corresponding kinematic space metric is If we consider the Poincar´epatch ofinstead the of bulk a AdS, its cylinder.entropy conformal ( boundary In will that be limit, a plane Σ we have rederived here isrelated also to universal parameters for of all a CFTs, holographically not dual just bulk holographic2.2 geometry ones by where eq. ( Poincar´epatch waist at geodesics: the entire expanding portionone of orientation, de while Sitter the (above contractingopposite region the (under waist) orientation. maps the The to waist) geodesics maps all a to cover the the Poincar´edisk (cf. same full figure geodesics constant but time with slice of AdS Figure 3 a Penrose diagram, using coordinatesembedded defined in in flat eqs. space ( JHEP09(2016)154 ). 18 (2.14) (2.15) (2.13) , and so 1 , depicted 2 − β ), and (b) the B.11 ) (2.16) , rather than angles,  ∞ , ) 2 is the horizon radius, ) u coordinate ranges over + ) and ( dφ ∞ r − 2 coordinates (dashed). The β φ r v θ < θ < ( B.10 lengths + π 2 −∞ dr is the planar patch of dS < u, v < , 1 sinh 3 − ∞ β  π −∞ 2 +  ( r α < 2 taken to be ]. We refer to the geometry as the “BTZ ` − log ≤ v 2 c 3 28 . The formula for the entanglement entropy r ,  (0 ] β , represented in (c) as the half-plane (cf. figure )  = is the AdS radius and 3 27 u [ and 25 ` – 8 – − +  β v u 2 ( /β is [ β π 2  2 ). πL dt  u dudv dθ β πα π`  2 2.3 2 − 2 + +  sinh r v 2 )–( = 2 2 2 sinh ` β − = π + 2 dα 2.2 2 r L  L 2 r − sinh ), but with 2 β 2  π log 4 1.1 . We discuss the BTZ black hole in the next subsection. L 2 dimension of length − 2 ), it is universal. π c 3 β π = = 4 ≤ 2.4 2 = 2 φ time = ds ds ent ] of length 2 ≤ S ds π u, v , and as the “BTZ black hole” when considering the quotient space, which − ∞ to φ . The kinematic space for the Poincar´epatch of AdS < φ < We again apply eq. ( The BTZ black string is dual to a CFT in a thermal state at temperature or equivalently in the coordinates defined in ( parameterizing the endpoints ofthat the the interval metric along on the kinematic infinite space spatial is boundary. We find of an interval [ and, similarly to eq. ( −∞ restricts effectively lives on acompact Euclidean-signature imaginary cylinder with an infinite space dimension and a is a nontrivial solution of 3drelated gravity, to where its temperature via black string” in the unwrapped (covering space) case, when the geodesics cover the full constant time slice of AdS 2.3 BTZ blackThe string BTZ metric, Figure 4 in red in (a) theembedding Penrose diagram, diagram, with using lines coordinates of defined constant in time eqs. (solid) ( and constant JHEP09(2016)154 3 ). ), θ, α 19 B.13 (2.18) (2.17) lengths ) and ( . ) B.12 lines (dashed). (c) ∞ θ < χ < . −∞ , πθ β ]. The boundary CFT effectively 2 ∞ 29 = , τ < 28 ≤ are the same, which is not necessarily the , χ L is temperature dependent in the BTZ case. β πα – 9 – L ) (0 2 and a compact imaginary time direction of length τdχ that R 2 isometry group [ 4 log tanh 3 ). The quotient results in space-like geodesics with new − = A + sinh τ 2 . Note that this is only equal to the hyperbolic patch of the AdS dτ 5 ), these coordinates and ranges cover the hyperbolic patch of de − ( 2 when the de Sitter radii B.5 L 2 coordinates along the compact spatial circle, related to the = 2 (see appendix ds + /r angular R . The kinematic space for the 1-sided BTZ black string is the hyperbolic patch of de Sitter, π` to be To parametrize these geodesics, we will again take the midpoint and opening angles The geodesics corresponding to the hyperbolic patch cover the region outside the hori- We can convert to hyperbolic coordinates via the transformation = 2 β global characteristics, including somecircular that horizon. wind an arbitrary number of times aroundθ, the α 2.4 BTZ blackThe hole BTZ black hole geometryappropriate can subgroup be of obtained the from AdS lives the on black a string by torus quotienting with by spatial an radius zon on the spatial BTZboundary. slice Due and to the are boundary homologouscorrespond being to to in CFT a geodesics intervals mixed on withcluded state, a complementary opposite in single intervals kinematic no orientation asymptotic longer space andcan once indeed, be we represented such as restrict intervals a towhich pure are state a amounts not (the to single thermofield in- considering boundary. double the state) Alternatively, 2-sided on the two BTZ copies state black of string. the CFT, By the embedding ( Sitter, depicted in figure kinematic space dS case. Indeed, we will see in section Now, the metric takes the form and (b) the embedding diagramThe geodesics with cover constant one time half lines of (solid) the and Poincar´edisk, that constant is, one outside-horizon region (cf. figure Figure 5 depicted in green in (a) the Penrose diagram, using coordinates defined in eqs. ( JHEP09(2016)154 ). of a 2.14 (2.19) π . , defined in ) ] for recent c ). The green α α 32 ]. This can be – + 2.22 in the Poincar´e θ 24 30 ≤ times winding when φ n ≤ φ < π , marked by black dashed α and ≤ − ). Depending on whether a π θ π ( φ < π 19 ≤ − ≤ α π for some example geodesics in the −  6 R β limit) and the non-winding geodesic for πθ ) with maximal opening angle 2 and figure − . The profile of a space-like BTZ geodesic is A → ∞ R R α < π β , the maximal extent it can reach into the bulk – 10 – α πφ has a width less than the angular width α 2 ). See figure R  = 1.). π β l 2 ∞ πα 2 ≤ and θ ) in the cosh n < π − cosh ≤ 2.19 ≤ R = 2 π β − πα R 2 . 2 β/ R β πα (with 0 2 cosh ) with winds around the singularity once and enters the entanglement shadow region. π r lines bounding the fundamental domain + θ, α coth r + 1) φ + α > π r n ( ) = . (a) Two example geodesics on the covering BTZ black string slice and (b) their coun- ) = ), and touches the entanglement shadow (in dashed red) defined in eq. ( α, θ α α < ; ( φ The geodesics that compute entanglement entropy are the minimal, homologous ones, In terms of the spatial slice we are considering, the quotient amounts to identifying 2.21 ( ≤ crit r r However, there existsgeodesics a that critical have smaller size lengthinterpreted past than as the which a connected “phase there homologousstudies. transition” is ones in [ The a the disconnected entanglement newgeodesic entropy geodesics (obtained family — consist from see of eq. of [ ( disconnected the disjoint union of a horizon-wrapping or infinitely winding. Thenπ geodesic is non-winding when covering space and its quotient. which we refer toTakayanagi as geodesics Ryu-Takayanagi geodesics. are the For non-winding sufficiently small geodesics intervals, whose the length Ryu- is given by eq. ( the constant disk representation of theBTZ slice geodesic (see appendix ( fundamental domain, crosses overthe multiple respective identifications images of or the coincides geodesics with in the the horizon, quotient space will be non-winding, winding For a given geodesic withis opening angle of the BTZ string byinherited a from factor the of associated theand minimal radius is parent given geodesic by in the BTZ black string geometry, boundaries. The blue geodesiceq. is ( non-winding ( geodesic with (The plots were made for Figure 6 terparts on the slicean of infinite the number quotiented of BTZ copies black of hole the geometry. fundamental The domain covering space consists of JHEP09(2016)154 u = (i.e., /β (2.20) (2.21) (2.22) . π R π  2 / θ 1  . /β , , ] with opening angle c R c v, u  R α > α 2 α < α β , π . 4 7  and midpoint angle  α  R ), the homologous, connected, non- exp 2 β R − π ) 1 2 ]. When the interval opening angle is 4 α π < π − β +  π u, v c ( 1 2 π 2  exp , the critical angle for the phase transition c < α 1 2 α log 2  sinh + 1 2 R π/ – 11 – β 1 2 π β  in the high-temperature limit πα   2 π coth log (with log ]. c 3 sinh c + R r 30 β β π + π ). This is illustrated in figure 4  v R ) = 2 β c = α > α π log 2 α c ( and c c 3 3 α the Ryu-Takayanagi geodesics cannot probe below the radius u crit    + r r = = = , which approaches ent r S max r < π 3 c ). 8 < α . Constant time slice with orange interval [ 2 but with the orientation reversed, so that together with the horizon-wrapping geodesic it is (the Hawking-Page phase transition temperature), below which the bulk geometry π/ (counterclockwise). The boundary-anchored portion of this disconnected geodesic is equal to The geometry exhibits a temperature-dependent entanglement shadow, since apart The critical angle is obtained from equating the contributions from each branch: The lengths of these geodesics compute the entanglement entropy of intervals, which α Such discontinuities arise from considering classical gravity in the bulk, and will be smoothed out when π corrections are taken into account [ v 3 2 − / /c The maximal entanglement1 shadow occurs for the lowest allowed temperature 1 with from the horizon with a discontinuity in the(see first figure derivative at is piecewise defined as homologous to the big orange interval instead. the complementary interval, which hasinterchanged opening endpoints angle winding geodesic in blue (with counterclockwisegeodesic orientation) in has larger green. length The thanhas the latter disconnected is counterclockwise homologous orientation to and theto the orange interval non-wrapping if boundary-anchored thethe geodesic horizon-wrapping minimal geodesic runs boundary-anchored from geodesicπ for the complementary interval [ Figure 7 larger than the critical angle JHEP09(2016)154 (2.24) (2.25) (2.26) (2.23) . The 6 1, when the . , The kinematic  ) c π 4 ≤ /β c R α α < α < α = ≤ c π < θ < π 0 α α . − , c c ]. , there is a phase transition and a ) c c 15 5 α α > α α α < α − < α < π, ]. It is illustrated in figure α ( 33 δ ], where the non-minimal geodesics are included R  ) 21 R , α β 2 π 16 β R results from the discontinuity in the derivative − β boundary-anchored, oriented geodesics in these – 12 – c π πα ( ) (0 α . 2 π all = dθ 14 + coth α R + ) R β 2 2 and α π log tanh − β 9 R π dα − ( β πα π − 2 2 = = log tanh tanh )( 2 2  τ τ α ( R β f π csch csch 2 of the interval. At the critical angle R        α 2 2 L β 2 ) = π α 4 ( )), it is not the space of f = 2 2.1 . The entanglement entropy of boundary intervals for the BTZ black hole, as a function ds The metric can be mapped to hyperbolic coordinates in the two finite regions, using As kinematic space is by definition constructed out of entanglement entropy data Here we take a different point of view than in [ One can check that the presence of the delta function is crucial for correctly computing the lengths of 4 5 in ‘kinematic space’.space To compare of the to covering their AdS, notion see of figures kinematicbulk space, curves it via suffices Crofton’s to formula consider as the a volume kinematic in kinematic space [ the redefinitions to hyperbolic time The delta function in theof metric the at entanglement entropy across the phase transition. where (see eq. ( quotiented geometries, butspace specifically is the given by: Ryu-Takayanagi geodesics. size, the entanglement entropy approaches the thermal entropy. is thermal AdSsmallest rather allowed than shadow region aRyu-Takayanagi occurs geodesics BTZ in can black the reach high-temperature hole all the limit [ way to the horizon. Figure 8 of the opening angle new family of disconnected Ryu-Takayanagi geodesics have minimal length. For very large interval JHEP09(2016)154 to lower ∞ (2.27) (2.28) ). The α = τ 2.23 boundaries, c α R ) c = α β ∞ − α π ( . π . In the second phase 10 2 < τ < R c β log tanh πα (in brown). The constant- in figure ) is arbitrary as far as the metric π 2 (not drawn) would separate the non- + 2 c < τ < 2.26 θ log tanh )–( is also reversed to undo the change in ∼ −∞ − θ . θ τ π > α ( R 2.25 β = π 2 α half of the hyperboloid. – 13 – = ) 2 χ in eqs. ( τ τ dχ half of the dS hyperboloid (as for the BTZ string). This 2 lower back , and the sign of upper π + sinh 2  dτ θ rotation, are glued together along their constant − π ( marks the maximum opening angle before minimal geodesics become non- → 2 + boundary at even larger θ c L θ α α = → = 2 θ boundaries are identified in the quotient, along with each subsequent boundary of and α ds θ α . (a) The Penrose diagram for the kinematic space of the covering BTZ black string, − π The sign of the time coordinate This covers two disjoint portions of the two hyperbolic patches, → = 0, or to cover part of the curves, forms aα subregion of theorientation upper of hyperbolic the patch geodesic.disconnected of Ryu-Takayanagi curves This dS past portion theof of phase the kinematic transition, hyperbolic space, is patch consisting in mapped the of to a the subregion family of resulting kinematic space is depicted as embedded in dS is concerned. It determinesthe the dS orientation hyperboloid of is theα covered. geodesics, or The equivalently, minus which signportion half is of of chosen kinematic to space, map which future consists infinity of the non-winding connected Ryu-Takayanagi which, after a with a defect along the identification corresponding to the delta function in eq. ( different constant- minimal non-winding geodesics fromdepicted the on the winding Poincar´edisk. ones. (b) An example geodesic forcombined with each the region angular redefinition either remain minimalwinding (green (dark and blue). brown),Its The become constant innermost non-minimal green(an (orange), region infinite is or number the of) become kinematicboundary fundamental infinitely space at domain of copies theminimal. BTZ The black non-minimal hole. region contains both non-winding and (finitely) winding geodesics: a Figure 9 color-coded to show the regions of geodesics that upon quotienting to the black hole geometry JHEP09(2016)154 is a sub- n singularities Z corresponding n 2 Z , where n Z / rotation are glued together 3 π , and in the high-temperature = 1) which gives the maximal + ] and start from the metric l /β θ 34 R → (and that no longer belong to kinematic θ . π 2 2 C / an integer. Conical singularities with an = 1 n /β ), and (b) the embedding diagram, with lines of R – 14 – B.13 ], see appendix 16 ) and ( coordinates (dashed). (The arrow in the embedding diagram . The Ryu-Takayanagi geodesics cover a region outside the θ 5 B.12 boundaries, with a defect along the identification. The portion of the c . α + r approaches the full upper hyperbolic patch that is the kinematic space of = = π α r 2 / 1  . The kinematic space for the BTZ black hole, shown in (a) the Penrose diagram, . For a comparison of our picture to existing work on the kinematic space of the /β 9 R Additional regions in the full hyperbolic patch of dS 2.5 Conical singularity We can obtain agroup conical of singularity the geometry spatial as rotationarbitrary a group deficit quotient angle SO(2) AdS are with alsoin solutions of this 3d paper. gravity, but we only We consider follow in this section the notation of [ space correspond tofigure geodesics that become2-sided winding quotiented upon BTZ black quotienting, hole as [ illustrated in entanglement shadow on (c) thedepicted covering here BTZ for black the string lowestentanglement allowed slice temperature shadow and (d) region the foroccurs quotiented in the BTZ slice, the BTZ high-temperatureup geometry. limit to the when The horizon the minimal Ryu-Takayanagi geodesics entanglement can shadow extend region all the way consists of two distinct subportionsto of the the two upper phases and of loweron Ryu-Takayanagi hyperbolic geodesics, their patches which of after constant dS a upper hyperbolic patch that islimit covered increases as athe function of BTZ string, depicted in figure Figure 10 using coordinates defined inconstant eqs. time ( (solid) andindicates constant that the lower patch is actually located on the reverse side of the hyperboloid.) The space JHEP09(2016)154 π ≤ θ (2.30) (2.31) (2.29) (2.32) (2.33) (2.34) ). The ≤ 2.38 π − = 4 copies of . n ) π ≤ ) have , φ 2 ˜ φ θ, α ≤ d 2), defined in ( 2 π π/ ], R − ( = + 36 , 2 max 35 dR 2 α 2, in order to be minimal. The green ( 1 and an infinite time-like direction, − dφ π/ crit 2 φ  r r = 2 , 2 ` + α R n π 2 , ) and ≤ ˜ dr φ , n 1 + 1)-winding geodesics ( n ˜ R 1 2 φ n nT .  − − = 4. The covering space consists of π/ ≤ = = =  k + – 15 – n 2 t 2 = r 2 n π φ ` r for some example geodesics in the covering space . − dT + max A 11 α  2 1 (˜ 2 n 2 ` R  crit R + geometry for 2 1 + ), marked by black dashed boundaries. The red dashed circles are the n  dt Z / −  . See figure 2.30 2 2 = ` r 2 + winds around the singularity once and enters the entanglement shadow region. ds 2 1 α < kπ n ) with a compact space-like direction  ≤ R α > π − π π (2 . (a) Two example geodesics on the covering AdS slice and (b) their counterparts on the = 1) C is the AdS radius, but covering only the restricted angular range 2 , − 3 ` ds k 1) times around the singularity at the origin, depending on how many times they cross In this geometry, parent geodesics of AdS get mapped to geodesics that wind up to Alternatively, we can change coordinates to − n ( the fundamental domain of theand quotient: ( ( and its quotient. The dual CFT livescylinder on the conformalfollowing boundary the of discussion this in metric, appendix which we take to be the This gives the standard metric for the conical singularity [ with the endpoints identified. of AdS where the fundamental domain ( respective entanglement shadows blue geodesic has the maximumgeodesic allowed with opening angle, Figure 11 slice of the quotiented AdS JHEP09(2016)154 ] ), ). )) ` u, v 2.4 2.34 (2.37) (2.38) (2.35) (2.39) (2.36) . , ) ) α π + . ≤ 2 the Ryu-Takayanagi θ θ π 2 π ≤ , , ≤ 2 φ ≤ 2 2 π π π π/ ≤ α < α < π/ − α < α = ≤ α < α > π, 2 π α 0 − . θ ≤ ( . n π α 2 n α
≤
α , but taken with the opposite orientation. 2 cot
π n − ) gives the metric n π α cot in the conical singularity geometry ( n α π ` n − 2 `  n α 1.1 θ α sin sin 2 it is equal to the corresponding curve for the cos ) = ) (0 R R – 16 – δ → ) =   2 − n n α n
α 2 2 θ
) max ( dθ n θ π α α 2 n
− n ( − + crit cos α > π/ φ n α π and ( r log log 2 crit c c 2 3 3 α r 2 2 cot dα − n ( cos − csc 2 csc π = )( q      α → ` n ( and opening angle ), and for ent g α θ S 2 2 ) = as a red dashed circle. n L ) = α 2.35 ( g 11 = α, θ . The entanglement shadow is defined by the minimal radius probed by 2 ). ; c φ ds ( 12 r We can imagine replacing the singularity with a small star with negligible back- The kinematic space prescription in eq. ( The entanglement entropies computed from these geodesics are given by (cf. eq. ( Since geodesics can be effectively deformed through the origin, for a given interval [ We will compute the entanglement entropy of a dual CFT interval from the length of The quotiented geodesics descend from the solutions in pure AdS (with AdS radius where It is shown in figure shown in figure Ryu-Takayanagi geodesics: curve for its complement The Ryu-Takayanagi curve is thedoes one with not minimal wrap length, around whichcurve the is is origin. always given the There by curve are eq.complementary that two ( interval, phases: with opposite for orientation. to considering a purebe state subleading on in the boundary (any entropy carried bythere the are star now is twothat homologous taken does boundary-anchored to not geodesics. wrap around These the consist origin of with a respect geodesic to the interval, as well as the analogous Star. reaction on the geodesics.there The is outside no geometry boundary isshare the to Ryu-Takayanagi same, spacetime curves with at while the the satisfying key origin. the difference homology that This condition. allows complementary This intervals corresponds to the corresponding geodesic, butof first holographic we entanglement need entropy. to We carefully distinguish two enforce cases: the homology condition Given a midpoint angle the radial profile of a geodesic is For a given opening angle, the maximum radial extent of the geodesic is JHEP09(2016)154 (2.41) (2.42) (2.40) rotation, the 2 boundaries. π . Such winding π/ , + 14 = θ n π ∞ 4 α → θ < t < 2, there is a phase transition . log tan n π/ π 4 ) ) π 2 π = t < c ≤ ≤ α α α log tan ≤ ≤ −∞ ≤ − 2 π (0 ( ( , θ n  = n 2 α α 2 – 17 – ˜ ˜ θ θ n − 2 π td . 2 that no longer belong to kinematic space correspond 2 , which we refer to as “sub-de Sitters”. The boundaries 2.4 log tan 13 − + cosh = = log tan ), the kinematic space covers the two different regions of the 2 t t dt B.3 ( −  2 L of the interval. At the critical angle α = ). The situation here is very similar to the effect of the phase transition in all cases. 2 n π ds 2.39 ≤ ˜ θ correspond to geodesics that are identified by the quotient, and so the vertical . The entanglement entropy of boundary intervals for the conical singularity, as a function ≤ n π , n π n π − − The regions in the full global dS From the embedding ( The metric maps to two subregions of global de Sitter space: In the finite regions, we can convert to global coordinates through the angular trans- = ˜ Along the identification, there isto a the defect metric corresponding ( tofor the the delta BTZ function black contribution hole, see section to geodesics that become winding upon quotienting, illustrated in figure θ boundaries are identifiedaccounts in for kinematic the geodesics space. pasthalf the of The phase transition, the subregion which hyperboloid are connectedtwo due located sub-de to to on Sitter their the past bottom regions reversed infinity (before back orientation. corresponding and to After after a the the phase two transition) families are of glued Ryu-Takayanagi together geodesics along their with global de Sitter depicted in figure The positive sign for the timetaking coordinate the has been opposite chosen orientation insince the of second a the phase flip to geodesic in account associated for orientation to maps to the the complementary opposite interval, portion of the de Sitter hyperboloid. combined with the time redefinition of the opening angle and a second family of complementary Ryu-Takayanagi geodesics have minimal length. formation Figure 12 JHEP09(2016)154 ≤ rotation, are glued together π + θ → θ coordinates (dashed). (The arrow in the θ ] to descend from a CFT concept called 34 – 18 – th fraction of the full de Sitter. In all cases the vertical n = 4 in (a) the Penrose diagram, and (b) the embedding dia- n For a true conical singularity geometry, the singularity is part 2 boundaries, with a defect placed along the identification. For a given , are conjectured in [ . This results in two phases for the entanglement entropy and correspond- π/ π = )) delineates the entanglement shadow region around the origin (red dashed). ≤ α α > π α 2.38 . The kinematic space for the conical singularity covers two “sub-de Sitter” patches, . Since there is no horizon, it seems there would be no transition to a pair of π ≤ , the entire purple region takes up an of the boundary of spacetime, andentropy the geodesics of that compute intervals the approaching holographicsingularity. entanglement the In entire other circle words, wouldα the be geodesics required would todisconnected have wrap an geodesics around opening (of the the angle kind in we the saw range for 0 the BTZ black hole). However, for several geodesics, with ‘entwinement’, associated with entanglement between internal (gauged)rather degrees than of position freedom, space entanglement entropy. True conical singularity. boundaries of these sub-decovering Sitter AdS patches space are coveredgeodesics identified. by (eq. (c) Ryu-Takayanagi ( The geodesics.(d) region The of The Ryu-Takayanagi geodesics the maximum cover spatial thethe radial slice spatial entanglement reach slices shadow. of of of the the these conical singularity geometry up to embedding diagram indicates thathyperboloid.) the For the lower star, patch geodesicsgeodesics is can satisfy be actually deformed located throughingly, on two the distinct origin, the portions and reverse the ofon side Ryu-Takayanagi kinematic their of space constant that, the aftern a Figure 13 depicted in purplegram, for with the lines case of constant time (black) and constant JHEP09(2016)154 , 2 0 ], α c → 38 , (2.43) ¯ A 37 S ] 40 from above. , 39 tot . Since S ¯ A S lower boundary marks are large, correspond- + α 0 A ¯ L , S  = 0 ≤ 0 ¯ L ¯ A L c 6 ∪ will approach A r S A π S , leads to a monotically increasing = π + 2 0 ≤ tot L S c 6 α : ¯ r A ≤ (in brown). The constant- π . In a number of recent works, e.g., [ 2 – 19 – 2 π  + 2 θ exp ∼ ≈ θ ) 0 ¯ L , 0 L ) in the range 0 that should either be replaced by a smooth geometry or dressed Ω( and its complement 2 2.37 conical singularity geometries have been identified as dual to CFT A 3 /CFT 3 . π . (a) The Penrose diagram for the kinematic space of the covering AdS space, color- fundamental domain copies approaches the entire system, we expect that n The second reason comes from CFT The first reason is that one generally expects the entanglement entropy of a subsystem A boundaries are identified in the quotient with each other, along with each subsequent boundary weight. We expect that this degeneracy is bounded by the Cardy formula [ where we are considering cases where the conformal weights asymptotically AdS states excited by thestates, insertion by of construction, and a are heavysmoothed interpreted out primary as by operator. dual the to presencestate a These of by geometries CFT a combining whose star, states many singularities as such are are wea pure pure studied states, mixed above. but state the One is maximal could roughly von construct log Neumann a Ω, entropy mixed where of such Ω is the density of such states at a given conformal Assuming no transition inthe the first geodesics, part i.e.,holographic of using entanglement eq. entropy the and ( entanglement oneapproaching entropy can from check that only this violates subadditivity for that we discussed above for the star. to decrease as the subsystemapplied approaches to the a total system system.as This comes from subadditivity reasons we are ledcontext to of interpret this AdS nakedin singularity a as possibly an unphysical Planck-scalesingle-interval idealization horizon. entanglement in entropy Either the and possibility the results same kinematic in space, the to same leading behavior order of in the remain minimal (purple andThe brown) innermost purple or region become isθ the non-minimal, kinematic i.e., space of wrapping theof or conical the singularity winding geometry. (orange). Its constant the maximum opening anglegeodesic before for minimal each geodesics region become depicted non-minimal on ones. the (b) Poincar´edisk. An example Figure 14 coded to show the regions of geodesics that upon quotienting to the conical singularity geometry JHEP09(2016)154 . c 2 proper in the large 2 / 1 c . 15 , these regions may form a 2 – 20 – subsets of dS ] indicates that adding a quantum field to the geometry 42 , 41 = 0 and has zero length. The resulting single-interval entanglement r . Different possible shapes for the Penrose diagram for kinematic space, for the Poincar´e of the domain of dependence of any Cauchy slice. They form globally hyperbolic It is also important to note that as For the quotiented BTZ black hole and conical singularity, the identification of Finally, recent work [ light ray emanating from athan point inside the the Cauchy subregion surface could and end the at spacetimes these would boundaries not rather subset be globally hyperbolic. regions only once they are considered as distinct spacetimes with boundary. geodesics along a boundarywe is have seen crucial that for the the Penroseregions diagrams causal for (with structure these curved examples of boundaries are kinematic in unionsin the space. of the convex case rectangular Indeed, of case the of BTZ phase black hole) transitions. that are Without glued the together identification of the vertical boundaries, a In this section weown want right. to In consider the these examplesproperty: we kinematic have spaces considered, they these as are spacetimes distinct sharesee globally spacetimes an this, hyperbolic important in the causal and their Penrose admitsurfaces diagrams Cauchy for for surfaces each the at are variousstring, schematically kinematic future BTZ depicted spaces infinity. black in as To hole, the well and case as conical of sample singularity the Cauchy in Poincar´epatch, BTZ figure black 3 Causal structure Up to this point, we have considered all the kinematic spaces we found as subsets of dS and including backreaction dresses the singularityleads with one a to Planck-scale expect horizon. a This transitiondiscussed picture in the for holographic entanglement the entropy very BTZ similarreduced to black to that hole, a point but at whereentropy is the then horizon-wrapping the geodesic same is as effectively we discussed above for the star geometry. ing to heavy states. Thislimit, would which lead is to subleading. a Hence, maximalsuch as entropy states far that are as goes effectively holographic like entanglement pure entropy states. is concerned, Figure 15 patch and BTZ blackFor string the (left), conical singularity a and conicalidentified. the singularity BTZ In (center), black all hole, cases and the thedomain the resulting time-like of BTZ and spacetimes dependence are space-like black is lines globally hole the hyperbolic, are (right). separately full with Cauchy spacetime surfaces depicted whose as curved red lines. JHEP09(2016)154 , the 3 For a given 6 , i.e., geodesics whose midpoint is θ or it could correspond to a maximum ]. − 45 I – or 43 , + I 29 – 21 – 2). On the space-like slice, these subgroups descend to , The various non-quotient space solutions of asymptotically ). 5 Additional solutions can be obtained as quotients by a subgroup and 4 ). In all the cases we know of, these regions are either unbounded or are A b). The geodesics on either of these vertical domain boundaries of kinematic a). Such regions are manifestly globally hyperbolic, with the future boundary isometry group SO(2 slice. For example, this could be 3 16 16 α gravity, restricted to a space-like slice, cover different regions of two-dimensional The quotient also introduces an additional subtlety due the possibility of phase tran- In constructing kinematic space, a geodesic need not remain confined in a single Examples of non-quotient space geometries include the Poincar´epatch and the BTZ The bulk regions will, in general, intersect the conformal boundary at a collection of We will now argue that this feature is a quite general property of kinematic space. 3 Since kinematic space is currently formulated for single intervals only, we disregard geodesics that con- 6 sitions, when the geodesic length may benect minimized disjoint boundary by regions, differentIt which would would classes correspond be to interesting of to the geodesics generalize entanglement entropy the in of proposal multiple to intervals. include these cases. of kinematic space consistaligned of with geodesics either with boundary constant (see endpoint figure of thespace fundamental are domain exactly in identified the underconstant- covering the space quotient. Anyopening remaining angle. boundary is a (space-like) plane into polygonal fundamentaldomains domains are with identified geodesic under boundaries. the quotient These [ fundamental fundamental domain, since thegeodesic segment part that of is the fully contained geodesic in that the exits domain. will The be time-like domain identified boundaries with a Quotient space case. of the AdS subgroups of the transformations. M¨obius as If Fuchsian we groups, consider only then discrete the subgroups, actions known of the subgroups tesselate the hyperbolic disk or kinematic space that intersects the(see space-like figure future infinity of theas ambient a de Cauchy Sitter surface. space black string, and both theirboundary kinematic spaces (see are figures future lightcones that intersect the future connected boundary interval, allcausal geodesics future of confined the to maximalkinematic geodesic anchor space that to connects boundaries this the are twopoint interval endpoints null, of are of corresponding the the in boundary interval. to the to The interval, geodesics and the that these maximal share boundaries geodesic. a intersect at single Thus, the end- point this corresponding portion maps to the filled-in forward light-cone in hyperbolic space and may(see be appendix depicted as regionsunions of of the fundamental Poincar´edisk or domains, upper which we half may plane assume aredisjoint bounded intervals, by each geodesics. corresponding to a disjoint region in kinematic space. We distinguish two cases, firstPoincar´epatch, for and non-quotient the space BTZ geometries black string), (suchas and as the second global for BTZ AdS quotient black space hole geometries or (such conicalNon-quotient singularity). space case. AdS JHEP09(2016)154 ]. 45 separating the α and any divergences should ∞ = c ). In both cases the kinematic space line, which for example could correspond α 15 )). In the examples we considered, time-like – 22 – 2.39 ) and ( lines in each region and the ability to propagate across the 2.23 θ corrections are taken into account. Thus, we will assume that /c . (a) To the left, an arbitrary subregion of the space-like slice that is bounded by geodesics which exhibit phase transitions (see figure 7 Examples of quotient space geometries include the conical singularity and BTZ black If the entanglement entropy or its derivative exhibits a kink across the phase transition, More topologically complex examples include the pair-of-pants wormholes considered in [ 7 hole, is topologically afuture cylinder boundary. which is globally hyperbolic with a Cauchy surface at the we expect that thesebe kinks regulated are when anpropagation 1 artifact through of setting thisidentification defect of is the constant possibleinterface, and kinematic space well-defined. will still In be globally this hyperbolic. case, due to the time slice corresponding to this critical angle. the metric mayappearing blow in up the along metricsand eqs. the ( space-like glued geodesics interface crossingnull (see geodesics this indicates for defect that instance the still causal the have structure is delta finite not length function significantly affected. and Additionally, the behavior of to a maximum opening angle. different regions of parametercontributions space. of In each distinct thismatic family situation space of there covered geodesics. is by a This each critical leads type to of different geodesic, patches which of are kine- glued together along a constant an endpoint, with thethe largest lightcone. contained (b) geodesic If (dashed) weto corresponding instead the to consider left, the a geodesics quotient point thatkinematic whose at exit fundamental space the the domain domain region tip is may boundaries the of the still shaded consist be boundary region of represented interval in all (represented kinematicare geodesics as space. identified. with dashed The a vertical time-like Any lines). midpoint space-like at boundary Due either to is endpoint the a of quotient, constant- these geodesics Figure 16 (semi-circles and vertical lines inlie the fully hyperbolic inside plane) this and region intersectsspace map the (right). to boundary. a Geodesics The forward that lightcone kinematic that boundaries ends are at the null future because boundary they in correspond kinematic to geodesics that share JHEP09(2016)154 (4.1) (4.2) (4.3) . The to the . Given tot ~ θ A ]: ρ is the dif- ¯ 47 A ent = tr δS ρ ), and the modular and center ρ α ) is a boundary-to-bulk log , ) ρ 4.1 0 ~ θ ( tr( 00 − T -dimensional CFT in flat space- ] for a CFT in the vacuum on ). Given a CFT in its vacuum 2 d has radius , = ) 20 2 ~ θ A mod   ent H 2 − ~ θ S ~ θ 0 − α d e ~ θ − 2 ( 0 α + ~ θ (tr 2 − 2  / 2 − α dα -dimensional de Sitter space in planar coordi- mod ). The ball 2 0 d − 1 H ~ θ α – 23 –  − 1 − 0 d 2 e 2 − can be derived by conformally mapping = ] provides a means for obtaining dynamics on an d , θ α L on a d = 20 A 2 A = ) is the energy density operator. ··· A Z 0 L 2 , planar ρ ~ θ 0 1 π ( d/ P ds 00 − t, θ on a constant time slice in a T = 2 = A 2 mod and m ] that the fraction in the integrand of eq. ( H 1 for the full system, the reduced density matrix is − 20 2 d θ is defined by tot ρ ··· mod + H 2 1 (with coordinates θ ]. Furthermore, the existence of Cauchy surfaces close to the future bound- 1 = − 46 2 ,d 1 ~ θ R It is observed in [ We begin by reviewing the original discussion in [ The global hyperbolicity of these kinematic spaces implies that they are always causally of a scalar field ofnates mass with the de Sitterference time between the coordinate entanglement given entropy by of a the slightly radius excited of state and the the sphere. entanglement If propagator has a modular Hamiltonian that is the generator of Rindler time translations [ where a density matrix entanglement entropy with the restHamiltonian of the system is state, the modular Hamiltonianhalf-line, for which has the Rindler wedge as its causal development region and consequently kinematic space. 4.1 Vacuum on aConsider a plane spherical region time time, the authors observe that afield de is Sitter boundary-to-bulk contained propagator inperturbations for the a of Klein-Gordon expression the entanglement for entropy the satisfy modular the de Hamiltonian, Sitter anda wave that equation. plane, consequently then proceed to several generalizations. In each case we provide a match to 4 Relation to auxiliaryThe de “auxiliary Sitter de Sitter”emergent de proposal Sitter [ spaceformal of field arbitrary theory. dimension While from reminiscentis the of less direct entanglement kinematic and entropy space not of in obviously equivalent: two a rather dimensions, con- than the deriving approach the metric for a static space- well behaved, i.e., theylations can [ admit dynamicalary fields means with that well-posedthe boundary interior. initial conditions value set formu- there determine the entire propagation within JHEP09(2016)154 8 ) = ρ | (4.4) , on a exc θ ρ ( . A point d S ) the differences . The spatial, ρ reduces to the ( R ent 17 For a CFT in the S . That is, − i ) ) 0 θ exc . ( to spherical regions on the ρ ( 2 d 00 , centered around d with radius T ent L α h S 1 is said to be in the time-like − R S = along the boundary of the ball = B × ent = 2 2 1 + I δS R m L ) is the same as the kinematic space 4.3 ) and with mod directly translates into containment relations ρH , ). The region of de Sitter that is covered is the d , or length – 24 – in auxiliary de Sitter dS tr( . Indeed, the mapping in figure α i − = 0 p 2.12 2 ) contained in a ball ent mod A ). Alternatively, the relative entropy can be written as H δS ρ ) . 2 exc . Each point in de Sitter corresponds to a ball in this time 4 ρ 0) boundary conditions set by log d ), combined with the first law of entanglement entropy, the m exc → − 4.1 ρ = tr( 2 α of dS i tr( ∇ , the auxiliary dS metric ( ( + − ) is a scalar field that solves the de Sitter wave equation in planar 1 , measures the distance along the periodic space-like dimension. We mod ) I 1 0 H exc R h θ a. α, θ δ ρ ( 2 . By eq. ( i log ent δS exc , with ). The causal structure of dS mod ρ ent . In 2 dimensions, the balls are intervals and this replicates the causal structure H 17 . The mapping between points h B δS δ ) = tr( i − ρ = | The authors argue for a visualization of the mapping between the CFT and the emer- This can be derived from the relative entropy between an excited state and a reference state, defined as mod exc 8 ent is at the tip of the lightcone that projects to the corresponding ball. ρ H ( i h S δ of the expectation valuereference of state. the Relative modularinfinitesimally entropy Hamiltonian small has excitations, and the this the leads property entanglement to that entropy the it with first is law respect of always to entanglement. positive, the and in the limiting case of 4.2 Vacuum on aConsider cylinder now the case ofangular a vacuum coordinate CFT on aconsider cylinder an interval ofconstant angular time slice. extent 2 Matching to kinematic spacevacuum on of a the plane Poincar´epatchmetric of for AdS. the Poincar´epatch ofplanar AdS patch ( depicted in figure between spherical regions:future a of ball of kinematic space discussed inmapping section in figure gent de Sitter space as follows.asymptotic boundary The constant time sliceslice of the via CFT is the taken intersection to(see be of figure the its future future lightcone with δS perturbation coordinates, with future ( Figure 17 boundary CFT. The conformalp boundary is identified with the asymptotic future of dS entropy of the reference vacuum state, the “first law of entanglement entropy” tells us that JHEP09(2016)154 θ )) near 4.2 (4.8) (4.9) (4.5) (4.6) (4.7) (4.10) (4.11) (4.12) ) we see )), which ): planar depicted in P 2.8 4.11 2.7 2 α , α , + ) is a length while + θ ) and ( θ 4.9 ) and ( . → → B.3 ) in ( 4.8 0 0 0 θ θ ( ] ), ( 00 to the circumference of the , given in eq. ( T 3 48 4.5 , L α . 47 , θ . α . cos )
, θ 2 2 − 00 ) cos ), covering the full dS
to note that From ( α 2 ) T dθ α 4 ) θ − ) can be recognized as a boundary-to- 2.7 . α α + − ) sin − θ 2 θ 0 . − 4.5 U to have the same behavior as global sin θ − θ − dα P 2 0 − R 0 θ − − θ ( Z ( cos( 0 X 0 global )), θ = 2 α – 25 – π ( 2 P cos( 2 = dθ O L ), and is given by [ L 2 B.4 L α = 2 P O sin α + 4.1 ) + θ − = θ = α ) + mod Z 2 α 2 − H ds θ R global − π P θ − 0 ). − θ = 2 ( 0 θ 2 L ( ) is the kinematic space of global AdS 4.11 − − mod 4.5 H = = 9 in eq. ( π global planar P P . The matching required fixing the de Sitter radius ) is an angle. 3 We can “normalize” the propagator To write down the expression for the boundary-to-bulk propagator on this global de We could again associate with this modular Hamiltonian an emergent auxiliary de The modular Hamiltonian for the interval can be obtained by applying a conformal To see this, we refer to the comment on notation on page 4.10 9 cylinder with the kinematicfigure space of globalcylinder AdS over ( in ( Matching to kinematicthat space the of modular global Hamiltonian indeed AdS. takes the form This confirms our ansatz of identifying the auxiliary dS metric for a vacuum CFT on a requiring us to set the limits of the interval: This can be subsequently transformedresults to in global coordinates (using ( Sitter space, it isin easiest embedding to coordinates write first the (using propagator ( on the planar patch (given by eq. ( Sitter space if thebulk fraction propagator. in the Motivated integrandspace by of the for ( equivalence the ofassociated Poincar´epatch the with of auxiliary eq. AdS, ( de we Sitter make and kinematic the ansatz that the auxiliary de Sitter transformation to the planar result ( JHEP09(2016)154 = α ) in + (4.13) (4.14) (4.15) (4.16) (4.17) θ 4.4 − ): 0 θ , with radius , the inverse = 1 2.16 E β x , on a constant S cyl θ , . × ] is ) vac . 1 . 0 00 50 2 θ T R ) ( 0 in [ θ − 00 ( x ) T 0 00 ) is ) θ θ T ( . ) + 0 of the entanglement entropy 4.5 i θ 00 θ β − T 0 − cyl β , D θ ent α ( . ( centered around π α π vac 2 00 δS 2  α T β πα h β dθ πα cos 2 β πα 2 π 2 ) can be recognized as a boundary-to- sinh = 2 measures the distance along the space- c cosh + − )  0 α 12 2 ) L α θ 2 − θ + 4.16 sinh = θ 2 sinh dα sin β − β πα − solves the de Sitter wave equation ( 2 0 − 0 sinh 2 θ θ 1 ( ] and the spatial coordinate 2 R π 0 2 – 26 – L ent 2 θ 50 2 [ − β as the hyperbolic patch of dS cosh cos( π δS 0 0 4 10 sinh = dθ 2.3 0 dθ 2 = α 2 α L α 2 dθ + α + θ − − α θ − θ ds α θ + Z ]. This θ = − Z 2 θ 2 β 49 Z [ R m β π 2 = 2 R c π = 2 = 2 i 24 mod − H mod mod H = H h ) is the kinematic space of the BTZ black string, given in eq. ( i δ cyl , = ). The corresponding mass is now fixed, as a consequence of fixing the de Sitter 4.16 vac 00 ent 2.7 T h . δS We could again associate with this modular Hamiltonian an emergent auxiliary de The modular Hamiltonian for the interval can be obtained by conformally mapping By the first law of entanglement, the perturbation . That is, the imaginary-time-like dimension is compact with periodicity u The relation between our spatial coordinate − 10 0 This metric was identified in section θ bulk propagator. Motivated by theand discussed kinematic equivalences spaces of for thewith AdS, auxiliary eq. we de ( Sitter make spaces the ansatz that the auxiliary de Sitter associated Sitter space if the fraction in the integrand of ( An equivalent expression that is more similar in form to eq. ( like dimension. We consider antime interval slice. of length the interval to the half-line,on for which the the Rindler modular wedge, Hamiltonian and is is just given the by boost generator 4.3 Thermal state We consider in thisβ/π section a CFT intemperature a of the thermal state. state The on spatial a coordinate cylinder with global coordinates, and as suchspace defines ( a local dynamic degreeradius, of to freedom on the kinematic of the interval associated withis small excitations given with by respect to the reference vacuum state JHEP09(2016)154 ) to L 4.18 (4.18) (4.21) (4.22) (4.23) (4.19) (4.20) ), ( , ) in hyper- E 4.15 4.4 α , + ] for an emergent thermal 00 θ T 20 From ( → − 0 ) 0 θ ( ), covering the hyperbolic . ] and [ . i 00 of the entanglement entropy ) T 16 θ , θ 2.16 , D −
0 ent ) β θ . 2 15 θ ( ) thermal 00 π − δS α 0 2 00 β T by considering the limit close to the θ h ( T − π β πα π ). 2 θ 2 c cosh 12 β πα − 2 . 0 − 4.20 − thermal thermal θ cosh sinh P ) of the de Sitter propagator in embedding ( π β P = β πα − 2 2 4.7 = 2 Z O 2 sinh π – 27 – β β πα π 2 L in eq. ( 2 solves the de Sitter wave equation ( − cosh ) + π ) of the planar propagator. For those to match, we = 2 α 2 L = ) to express it in hyperbolic coordinates: ent cosh 4.9 − 2 = 0 2 δS θ mod L . The matching required fixing the de Sitter radius 2.17 dθ H − − 5 0 α α θ + = ( θ − thermal 2 θ π β P Z ]. This ) and ( m β 49 −L [ B.5 2 = = 2 β πc i 6 = mod i thermal depicted in figure H h P 2 ). The corresponding mass is again fixed, as a consequence of fixing the de δ ) we see that the modular Hamiltonian indeed takes the form = thermal 00 2.16 T ent h 4.20 By the first law of entanglement, the perturbation We can again “normalize” the propagator To write down the expression for the boundary-to-bulk propagator on the hyperbolic δS 5 Discussion and further5.1 directions Refined prescriptions We have shown that thede two Sitter distinct space prescriptions give from equivalent results [ in the case of a CFT on a plane, on a cylinder with bolic coordinates, and asspace such defines ( a local dynamicSitter degree radius, of to freedom on the kinematic with cylinder with the kinematicpatch space of of dS the BTZthe black circumference string of ( the cylinder over of the interval associated withis small excitations given with by respect to the reference thermal state and ( This confirms our ansatz of identifying the auxiliary dS metric for a thermal CFT on a fix the kinematic space de Sitter radius to Matching to kinematic space of the BTZ black string. boundary of the interval, and comparing it to the behavior ( patch of de Sitter,coordinates we and use eqs. the ( expression ( JHEP09(2016)154 ) 4.2 (5.1) (5.2) in ( , defined We remark mod planar H 11 P ): /π . on the kinematic at a constant time S 00 1.1 1 when the compact A = T − L ∼ 2 m with , times ] have argued that the MERA π c 12 16 , . 15 00 dudv ) T P u, v and a modular Hamiltonian ( A Z on an emergent de Sitter space with radius A ent ρ ∂u∂v π 2 S 2 – 28 – L ∂ / = 2 2 2 L − of a scalar field with mass mod c 12 = P H 2 = , and the integrand defines a boundary-to-bulk propagator m 00 T . However, this would not lead to the same matching between kinematic , the mass squared of the scalar is proportional to the energy ). As we have seen, the modular Hamiltonian can be written as S P L that depends on the length of the compact direction in the CFT. ], the UV cutoff scale of the CFT can be considered a natural length scale for A mod R 23 L H .) π − L e is the circumference of the cylinder’s compact dimension. = 2 (tr of CFT on cylinder S / 2 K mod ds mod H H − . Here e = /π S A We can also reverse the argument: knowing the entanglement entropies of closed in- By equating these two emergent de Sitter geometries, we obtain a dynamical scalar The emergent de Sitter space associated with the conformal boundary can then be Consider a (1+1)-dimensional CFT on a cylindrical conformal boundary, dual to either In some contexts, e.g. [ ρ of a scalar field with mass = 11 The refined prescription inCFT the lives previous on subsection applies a specifically cylinder to that cases was where obtainedthe the from kinematic a space, instead conformalspaces of mapping and modular of Hamiltonians the that we plane emphasize with here. For global AdS or theknown BTZ results, black but string, it we would were be able interesting to to5.2 check go this beyond line this. of Beyond reasoning universality with tervals allows you to calculateThe boundary-to-bulk the propagator kinematic spacespace of can the then constant be time used slice to in write the down bulk. an expression for the modular Hamiltonian: field moving on the kinematictensor space. network is The a authorsinteresting of discretization new [ of ingredient kinematic in space. the study Our of results this thus MERA-kinematic offer space a connection. potentially identified with the kinematic spacetime or slice space of the of bulk Ryu-Takayanagi geodesics geometry, on obtained the through constant the prescription in ( P L that with this choicedensity for of the CFTcoordinate with is proportionality imaginary factor time. equal to does not depend on (2+1)-dimensional global AdS or thein BTZ the black string. CFT has Each interval aby reduced density operator an integral, latter two cases are holographically dual toBased global on AdS these or examples the we (slightly) BTZ reformulate blackthat the string, they prescriptions respectively. for lead the to cylinder the cases,of so same the emergent de dS Sitter geometry. radius (For In the particular, CFT we on include a a plane specification there is no length scale present. Correspondingly, a compact spatial direction or on a cylinder with a compact imaginary time direction. The JHEP09(2016)154 ], ). R ], or ), in 13 (5.3) π, π α 38 − [ + ∈ θ 0 → θ, θ 0 θ and R , , we find some suggestive ], and if we normalize by con 00 4 T ). 20 we calculated the kinematic n α . 5.3 2 ) with the angles replaced by 2.5 the kinematic space of the BTZ cos R 2 n α − n , as in [ 4.16 2.4 θ 2 2 sin / − n L 0 1 / θ ]). The same is true for the entanglement 2 − − 51 = cos 0 2 = subgroup of the spatial rotation group SO(2). dθ m 2 α – 29 – n m α + Z θ − θ and Z 2 R , the radius of the spatial circle is ). This would suggest an expression for the modular n α R πn = 2 2.23 2. The fraction in the integrand has the form of a boundary- L = 2 ) by a bulk geometry. In section c con mod α < π/ H ]. It would be interesting to calculate the spectrum of eigenvalues of ], and use this to check or modify eq. ( ), we find 38 53 , , 4.9 37 52 ), suggests that for holographic CFTs the modular Hamiltonian of sufficiently 2.39 We do not know of a CFT calculation of the modular Hamiltonian of an interval in Similarly, one could write down the boundary-to-bulk propagator on the kinematic The kinematic space we found for the conical singularity, with metric given by In the realm of holographic two-dimensional CFTs, we can implicitly define a CFT The partition function of a CFT on a spacetime with a genus higher than zero is not space of the BTZHamiltonian of black an hole interval on ( aFor sufficiently spatial small circle at intervals, finite ittimes temperature is the (of just a angles. given holographic by For CFT). larger eq.transition intervals, ( in we the would have Ryu-Takayanagi to curves. determine the As effects for of the the conical phase singularity case, we do not a conical singularity state.certain limits [ However, thethe entropies R´enyi reduced for density such operatortechniques states (the of are [ entanglement spectrum) known in from these results, using the hold for intervals with to-bulk propagator for aIf scalar we field take on the theexamining mass sub-de the of Sitter behavior the kinematic of fieldanalogy space the to to (see eq. obey propagator figure ( near the end of the interval ( where the length of the intervaland is 2 we are consideringBecause a of quotient the by phase a transition in the entanglement entropy, we would only expect this to perhaps a statistical mixture of such states. eq. ( small intervals in such a state is of the form: state (to leading order in space of a conical singularityblack spacetime hole. and in The section corresponds latter to is a dual spacetime toto with a a the CFT CFT topology on on of a a spatial a spatial circle torus. circle excited The at by the conical finite insertion singularity temperature, of is which a dual heavy primary operator [ entropies in generic excitedintervals are states. similarly non-universal We inexplore expect these the cases, that two and emergent the de they Sitter modular maywe space do be Hamiltonians prescriptions not non-local in of find as two precise well. single suchresults matches, cases and We as below, avenues we and for did though further in the investigation. cases in section and so are universal for all CFTs. Doesuniversal, the i.e., equivalence it extend depends beyond on this? central the charge. full Consequently, spectrum the ofsuch entanglement operators spacetimes entropies are of of also the not intervals universal CFT in (see, and e.g., states [ not defined just on on its no operator insertions. The results in these cases depend only on the conformal symmetry JHEP09(2016)154 . 3 (5.4) (A.1) (A.2) ], and one through a 57 – ent S 54 2), the isometry , , 32 is the 00 component , 00 31 T . 2 2 , where ` . dY 00 − T + = 2 P 2 A everywhere. Globally distinct solutions Y R dX 3    y π + + 2 2 = 2 X , the modular Hamiltonians were already known auxiliary dS dV Lorentz transformations SO(2 + – 30 – and the auxiliary de Sitter space: 4 the AdS radius. The induced metric is 2 − mod 2 , K ` 2 2 V H −→ K ←− R − dU 2 ent − mod U S , with H = 2 , − 2 2 R ds of a scalar field on the kinematic space to the modular Hamiltonian metrics correspond to different classes of solutions of classical, pure P 3 through the formula ], which allows one to compute certain boundary modular Hamiltonians ) is left invariant by A 58 A.1 gravity. Indeed, (2 + 1)-dimensional classical gravity with a negative cosmological is defined as the locus How generally can one go from the kinematic space, which is relatively easy to calculate, 3 3 The resulting AdS AdS constant (and no sourcenegative curvature: terms) all is solutionssuch trivial are as in locally the AdS the BTZ black sense hole that and the the conical solutions geometry have are obtained constant as quotients of AdS The locus ( group of AdS.coordinate Classifying these systems isometries can into be orbits introduced in that space-like make and different time-like classes planes, of isometries manifest. A 3d gravity AdS in the flat embedding space recent work [ from bulk data usingalong relative with entropy. the relationship Thishigher between dimensions would kinematic and be space in interesting and time-dependent to the states. auxiliary investigate de further, Sitter space in density operator, which, aentropy. priori, In has the cases much considered more infrom information section CFT than just calculations, the soWe entanglement the expect sketched this loop toholographic served CFTs work as in only a many for states check dual rather certain to than CFTs classical a bulk in geometries. prediction. certain This states, is consistent but with this might include This would be remarkable since the modular Hamiltonian of an interval is equivalent to its of some region of the energy-momentum tensor operatorfrom the in asymptotic the behavior given CFTmodular of state the Hamiltonian (this bulk would can metric, be beloop in obtained that determined holographic includes cases). from the the In kinematic this entanglement space sense, entropy the have been extensively studied andcould are again known check in the a consistency variety of of limits the [ entanglement spectra. to the modular Hamiltonian?to-bulk The propagator general procedure would be to go from the boundary- know of a CFT calculation of the modular Hamiltonians. However, the entropies R´enyi JHEP09(2016)154 ]. ). π 2 (A.8) (A.3) (A.4) (A.5) (A.6) (A.7) , [0 A.11 ∈ φ , , 0 and lines (black dashed). (BTZ) φ 2 (AdS) r > instead of the unit radius dφ , consistent with ( 2 r 2 R R 2 + dφ , 2 dφ 2 r 2      r      dr ) ) ) ) + 1 + 2 − t/` t/` 2 t/` t/`  dr 1 dr 1 φ φ φ φ 1 sin( cos( − − 2 − 2 sinh( cosh(  , ` sin 2 ` cos 2 2 2  ` r 2 + r ` sinh r ` cosh + ` + r + r r  r to a circle of radius 2 2 − 2 − + 1 φ ` − r r 2 + 2 2 2 2 – 31 – r , represented as a Poincar´edisk on the left and the r √ √ 2 ` r (in solid black) and constant r → 3 √ √      dt  r → ∞  form:      r  + = + 3 1 , φ 2 = 2 of AdS      + − ` ). The radius of the disk is then dt dt r      2 U Y V X 2 2 t   ` r H U Y V X      2 + A.12  →      r + 1 2 ] in the universal covering space, radius − ` 2 2 − in ( ` r ∞ i 2 = , t , y r  2 + `  r − ds r −∞ − → R [ = i → = 2 ∈ y ) takes the pure AdS t 2 r ds ds A.2 . Constant time slice In ‘hyperbolic or Schwarzschild coordinates’ on the other hand, In ‘static AdS coordinates’ or, after the transformation the metric becomes the metric ( with AdS time Figure 18 half-plane on the right,We can showing imagine constant mappingPoincare the disk, boundary i.e. JHEP09(2016)154 ) ) + β ( /r A.4 C ) will R (A.13) (A.14) (A.10) (A.11) π` . Below A.8 ` : = 2  β + r . ) and ( → ∞ R r A.4 , . ) (A.9) 2 = 0) , . = 0) (A.12) t dφ t ( 2 2 ! 2 . When referring to this metric ( → ∞ 2 U R dy r U ∞ 2 dφ + ` + + 2 + `Y . For an unwrapped angle (covering . After Wick rotation to Euclidean . For AdS, the conformal boundary 2 2 1 ` R X R  t/` dy ). Both representations are presented in t + )( = = 2 2 ` 2 < φ < 2 2 R = 2 , is completely arbitrary from the boundary dy  dφ  2 t 2 → R the dominant saddle-point of the gravity path ` d + ` , so this is the 1-sided BTZ black string (it has R −∞ R , y t , y 2 1 π / − + ) (for the BTZ black hole), with – 32 – . It can be represented as a Poincar´edisk, with + π < φ < π 2 2  2 r / dy 2 = d ` − , β ( 1 = 0) 2 dt 2 2 = 0) − − R ` t y r > r t ) with radius ( − π

( ( = ). R = 2 U (2 2 2 2 2 U /β < π ` r r 2 R T + Y `X R 2 H + (2 = 0 of the AdS and BTZ geometries ( 2 conformal time slice `Y ` ) is conformal to the Poincar´edisk → = C + → ∞ t ds 2 2 ds X 2 2 conformal bdy 2 = r dy 1 = ds X ds y ) have the same behaviour near the boundary + 1 + y 2 1 , we refer to the spacetime as the BTZ black string. The metric 2 A.8 ∞ dy U ( to the unit circle. More precisely, the metric of a constant time slice 2 − ) 2 2 , for AdS and BTZ. y + ) and ( 4 19 2 1 the conformal boundary of the BTZ black hole approaches a cylinder rather slice ` y < φ < 4 → ∞ 2 + π A.4 2 r 2 ` H and / ) with rescaled time coordinates − 1 ( −∞ 18 A.8 =  Another possible representation uses the half-plane coordinates A constant time slice at The CFTs can then be said to live at the conformal boundaries of the bulks ( Both ( 2 2 H /β in which the metric becomes figures that map ds define an Poincar´ecoordinates (for the BTZ string)the or inverse a torus temperatureR of the blackthan hole a in torus, which thethe corresponds Hawking-Page bulk. temperature to considering In aintegral is macroscopic the no black high-temperature longer hole the limit BTZ black hole but rather thermal AdS. CFT point of view, hence the introduction of theand arbitrary ( length scale takes the form ofsignature, a the cylinder topology of the BTZ conformal boundary becomes either a cylinder conformal to The conformal factor that was dropped, as a BTZ blackrestricted hole, to it the is fundamentalspace), generally domain: assumed to becovers the the region quotient outside the space,one horizon with conformal coordinates boundary at where the range of the hyperbolic angle is JHEP09(2016)154 and (B.5) (B.1) (B.2) (B.3) (B.4) . ]. 2.3 2.4 60 planar patch is introduced α ] and [ in section cover the region 59 ∞ b): ≤ ∞ 5 cover the full de Sitter lines (dashed). The blue cover the ) of the 1-sided BTZ. The χ π φ < φ < 2 ∞ A.8 ≤ (figure −∞ θ . −∞ ≤ . . . , ≤ 2 < θ <    2       , 0 χ χ 2 2 θ θ L θ θ dY ≤ ∞ α α − − L L t τ = τ 2 2 −∞ + 2 2 sin cos ≤ ∞ , sinh α α cosh 2 t θ 2 t α 0 ≤ t τ τ Y + − sinh cosh dX + α α L L cosh 2 2 cosh hyperbolic patch 2 + sinh sinh α >       2 X – 33 – , covered by the metric (    the dS radius. The induced metric is −∞ ≤ L (solid black) and constant L + L + dU L r 2 = ). = − = U ). Another time-like global coordinate r > r       =    of BTZ, represented as a Poincar´edisk on the left and the − 2.12 b): U Y 2 X 2.9 U Y X ): 2 U Y ) with ranges 0 X , with corresponding to a wormhole-like geometry in section 4 2       ds H ,    1 τ, χ ) with ranges ≤ ∞ R 0, known as the ) with ranges α, θ t, θ π < φ < π , U > X,Y,U − ≤ ∞ ) takes the form ( ) takes the form ( Y −∞ ≤ B.2 B.2 . Constant time slice 0 (illustrated in figure is defined as the locus −∞ ≤ ≥ 2 , ). L U Hyperbolic coordinates ( Planar coordinates ( Global coordinates ( dS 2.8 + X > X The metric ( The metric ( in ( hyperboloid ( Useful references for the various coordinates on de Sitter space include [ in the flat embedding space maximally extended or 2-sidedto BTZ include is the obtained yellow region. whenthe the We quotiented consider metric metric the is unquotiented continued metric beyond the horizon B De Sitter embeddings and penrose transformations Figure 19 half-plane on the right,region showing is constant the region outside the horizon JHEP09(2016)154 . 12 4 2 P ]. dθ (B.6) (B.7) (B.8) (B.9) 61 (B.10) (B.11) (B.12) (B.13) , + 7 are used 2 P are intro- θ dt θ − black hole was . and = , 3 and 2 P   α θ ds , πθ β β πα ) 2 2 , and α sinh sinh Penrose diagram is rectangu- β πα < α < π πθ β and 2 2 in global coordinates, figure 2 . , P t 13 2 2 sech α α ), conformal to (for 0 sech 2 that gets identified. P  ,  − + and π dθ 2 2 θ α Y 3 X θ θ U, 2 2 ], but we will assume it is basically correct and + α to + − 64 2 P – − π 2 2 ) ) 62 dt L L − 2 π − = arctan ) in hyperbolic coordinates) are obtained through = arccot – 34 – 2.17 2.17 ( ( )( = arctan = arctan = 2.8 ) ( ) 10 P ). Other hyperbolic coordinates t P P χ χ ) ), the coordinates t θ 2 t and = arctan = arctan 2.18 B.9 sinh 5 cosh P P τ t θ τ )–( = (sec 2 B.8 ds θ , = arctan(sinh = P P t θ = arctan(sinh = arctan(tanh ) takes the form ( ] and depicted in their figure 17. There, kinematic space is referred to as the ) becomes P P ). t θ 16 B.2 the azimuthal angle in a range 2.9 constant time-slice geodesics, including the winding ones and a set of horizon- P 2.17 θ all In hyperbolic coordinates In planar coordinates In global coordinates The corresponding Penrose diagrams (figures There is some debate about this picture, e.g., [ 12 space of crossing geodesics which have one endpointtake on the each term of “kinematic the space” two to asymptotic regions, refer while to we the space ofexplore Ryu-Takayanagi geodesics some only implications, (in without taking a firm stance on the extent of its validity. C Two-sided BTZ blackMaximally hole extended, asymptoticallythese AdS can be black identified holes with twoThe boundary have kinematic CFTs two space in an associated asymptotic entangled, with thermofielddiscussed regions such state in a [ and [ two-sided, asymptotically AdS interchangeably when working in globalAdS coordinates, and conical namely geometry in kinematic thelar, spaces. with sections The on global the dS global the metric ( Note that because of eqs. ( in the respective embedding coordinates. For completeness they are explicitly given below. duced in ( in planar coordinates, and figures the Penrose transformations The metric ( JHEP09(2016)154 . , and 20 α ]. ]. This is done in figure 16 refer to hyperbolic coordinates, see whose hyperbolic patch is identified SPIRE θ 2 IN ][ and together have to be used to extend the α ] (right). We have shown in thick black lines 9 16 – 35 – hep-th/0603001 [ ), which permits any use, distribution and reproduction in Holographic derivation of entanglement entropy from AdS/CFT (2006) 181602 CC-BY 4.0 96 This article is distributed under the terms of the Creative Commons . The space of all constant time-slice geodesics of the two-sided BTZ black hole (left), ). Phys. Rev. Lett. 2.17 S. Ryu and T. Takayanagi, on the right are the global de Sitter coordinates of the dS [1] References glement Workshop at thework University and of learn Pennsylvania about for related the developments. opportunityOpen to discuss Access. this Attribution License ( any medium, provided the original author(s) and source are credited. Department of Energy (DOE) underNational DOE Science grant Foundation DE-SC0011941, of N.C. Belgiumsupported is (FWO) supported Odysseus by by grant NASA the G.0.E52.14N, ATP andInstitute grant C.Z. Visiting NNX16AB27G. is Graduate C.Z. Fellow program wouldtensor where networks, like she and to learned both thank about C.Z. the and kinematic N.C. space Perimeter would and like to thank the New Frontiers in Entan- We thank Dionysios Anninos,Boer, Alice Netta Bernamonti, Engelhardt, ChrisLin, Brust, Federico and Galli, Bartlomiej Herman Czech, Kurt Verlinde JanA. Hinterbichler, for de Rosen helpful Aitor for discussions Lewkowycz, discussions and Jennifer and input. initial We collaboration. especially thank C.A. Rachel is supported in part by the U.S. a one-to-one mapping betweenterminology, CFT all intervals the and colored bulk regionspicture geodesics). in to our the When figure two-sided following case their and compare to the figureAcknowledgments in [ (on the left) the boundaryθ of the domain depicted inwith the the figure BTZ on the blackwhile right. string in The kinematic the coordinates space. resteq. of They ( the are BTZ discussions used in only the in paper, this instance for comparison, Figure 20 compared with the depiction of it in figure 17 in [ JHEP09(2016)154 , ]. 42 ]. 06 , ] ] ]. (2006) JHEP ]. SPIRE , 08 IN SPIRE JHEP ][ SPIRE IN Bulk curves , (2012) 065007 IN ][ SPIRE ][ IN Gravitation from JHEP , , ][ Gen. Rel. Grav. Class. Quant. Grav. D 86 , , ]. arXiv:1208.3469 arXiv:1510.02367 arXiv:1310.4204 [ , [ (2003) 021 (2016) 061602 Casting Shadows on SPIRE arXiv:1412.5175 04 Phys. Rev. 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