Causality & holographic entanglement entropy
Matthew Headrick, Veronika E. Hubeny, Albion Lawrence, Mukund Rangamani
PPrepared for submission to JHEP
DCPT-14/33, BRX-TH-6284
Causality & holographic entanglement entropy
Matthew Headrick a , Veronika E. Hubeny b , Albion Lawrence a ,Mukund Rangamani ba Martin Fisher School of Physics, Brandeis University,MS 057, 415 South Street, Waltham, MA 02454, USA. b Centre for Particle Theory & Department of Mathematical Sciences,Science Laboratories, South Road, Durham DH1 3LE, UK.
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We identify conditions for the entanglement entropy as a function ofspatial region to be compatible with causality in an arbitrary relativistic quantum fieldtheory. We then prove that the covariant holographic entanglement entropy prescrip-tion (which relates entanglement entropy of a given spatial region on the boundary tothe area of a certain extremal surface in the bulk) obeys these conditions, as long as thebulk obeys the null energy condition. While necessary for the validity of the prescrip-tion, this consistency requirement is quite nontrivial from the bulk standpoint, andtherefore provides important additional evidence for the prescription. In the process,we introduce a codimension-zero bulk region, named the entanglement wedge, naturallyassociated with the given boundary spatial region. We propose that the entanglementwedge is the most natural bulk region corresponding to the boundary reduced densitymatrix.
Keywords:
AdS-CFT correspondence, Entanglement entropy a r X i v : . [ h e p - t h ] N ov ontents One of the remarkable features of the holographic AdS/CFT correspondence is thegeometrization of quantum-field-theoretic concepts. While certain aspects of recastingfield-theory quantities into geometric notions have been ingrained in our thought, weare yet to fully come to grips with new associations between QFT and bulk geometry.A case in point is the fascinating connection of quantum entanglement and spacetimegeometry. The genesis of this intricate and potentially deep connection harks backto the observation of Ryu-Takayanagi (RT) [1, 2] and subsequent covariant general-ization by Hubeny-Rangamani-Takayanagi (HRT) [3] that the entanglement entropyof a quantum field theory is holographically computed by the area of a particular ex-tremal surface in the bulk. In recent years, much effort has been expended in trying to– 1 –esh out the physical implications of these constructions and in promoting the geom-etry/entanglement connection to a deeper level [4–7] which can be summarized rathersuccinctly in terms of the simple phrases “entanglement builds bridges” and “ER =EPR”. Whilst any connection between entanglement and geometry is indeed remark-able, further progress is contingent on the accuracy and robustness of this entry in theholographic dictionary. Let us therefore take stock of the status quo. The RT proposal is valid for static states of a holographic field theory, which allowsone to restrict attention to a single time slice ˜Σ in the bulk spacetime M . The entan-glement entropy of a region A on the corresponding Cauchy slice Σ of the boundaryspacetime B is computed by the area of a certain bulk minimal surface which lies on ˜Σ.In this case we have a lot of confidence in this entry to the AdS/CFT dictionary; firstlythe RT formula obeys rather non-trivial general properties of entanglement entropiessuch as strong subadditivity [8–10], and secondly a general argument has been givenfor it in the context of Euclidean quantum gravity [11].However, it should be clear from the outset that restricting oneself to static statesis overly limiting. Not only is the field theory notion of entanglement entropy valid ina broader, time-dependent, context, but more importantly, one cannot hope to infer allpossible constraints on the holographic map without considering time dependence.The HRT proposal, which generalizes the RT construction to arbitrary time-de-pendent configurations by promoting a minimal surface on ˜Σ to an extremal surface E A in M , allows one to confront geometric questions in complete generality. However, thisproposal has passed far fewer checks, and an argument deriving it from first principlesis still lacking. This presents a compelling opportunity to test the construction againstfield-theory expectations and see how it holds up. Since the new ingredient in HRTis time-dependence, the crucial property to check is causality. The present discussiontherefore focuses on verifying that the HRT prescription is consistent with field-theorycausality . Let us start by considering the implications of CFT causality on entanglemententropy, in order to extract the corresponding requirements to be upheld by its putativebulk dual. As we will explain in detail in §
2, there are two such requirements. First,the entanglement entropy is a so-called wedge observable . This means that two spatialregions A , A (cid:48) that share the same domain of dependence, D [ A ] = D [ A (cid:48) ], have the sameentanglement entropy, S A = S A (cid:48) ; this follows from the fact that the corresponding We will focus exclusively on local QFTs with conformal UV fixed points which are holographicallydual to asymptotically AdS spacetimes in two-derivative theories of gravity. As we elaborate in the course of our discussion this result follows from Theorem 6 of [12]. Asthis is however not widely appreciated we focus on proving the result from a different perspectivehighlighting certain novel bulk constructs in the process. – 2 –educed density matrices ρ A , ρ A (cid:48) are unitarily related [13]. Second, fixing the initialstate, a perturbation to the Hamiltonian with support contained entirely inside D [ A ] ∪ D [ A c ] (where A c is the complement of A on a Cauchy slice) cannot affect S A . Thereason is that we can choose a Cauchy slice Σ (cid:48) that lies to the past of the supportand contains a region A (cid:48) with D [ A (cid:48) ] = D [ A ]; since the perturbation cannot change thestate on Σ (cid:48) , it cannot affect S A (cid:48) , which by the previous requirement equals S A . Time-reversing the argument shows that, similarly, S A cannot be affected by a perturbationin D [ A ] ∪ D [ A c ] when we consider time evolution toward the past with a fixed finalstate.Having specified the implications of causality for the entanglement entropy in thefield theory, let us now translate them into requirements on its holographic dual. First,in order to ensure that the HRT formula in general gives the same entanglement entropyfor A and A (cid:48) , they should have the same extremal surface, E A = E A (cid:48) . Second, in orderfor E A to be safe from influence by perturbations of the boundary Hamiltonian in D [ A ]and D [ A c ] (when evolving either toward the future or toward the past), it has to becausally disconnected from those two regions. This means that the extremal surfacehas to lie in a region which we dub the causal shadow , denoted by Q ∂ A and defined in(2.7) as the set of bulk points which are spacelike-separated from D [ A ] ∪ D [ A c ].This causality requirement takes an interesting guise in the case where A is anentire Cauchy slice for a boundary. If this is the only boundary, and the bulk is causallytrivial, then there is no causal shadow; indeed, E A = ∅ , corresponding to the fact thatthe entanglement entropy of the full system vanishes in a pure state. However, if thestate is not pure, the bulk geometry is causally nontrivial: typically the bulk black-holespacetime has two boundaries, dual to two field theories in an entangled state (whichcan be thought of as purifying the thermal state of the theory on one boundary). If wetake the region A to be a Cauchy slice for one boundary and A c a Cauchy slice for theother, then the extremal surface whose area, according to HRT, measures the amountof entanglement between the two field theories must lie in a region out of causal contactwith either boundary. How trivial or expected is the claim that the extremal surface resides in the causalshadow? It is interesting to note that for local
CFT observables, analogous causal-ity violation is in fact disallowed by the gravitational time-delay theorem of Gao andWald [14]. This theorem, which assumes that the bulk satisfies the null energy con-dition, implies that a signal from one boundary point to another cannot propagatefaster through the bulk than along the boundary, ensuring that bulk causality respects For the well-known eternal static Schwarzschild-AdS case, the shadow region degenerates to thebifurcation surface, but we will see that in general it is a finite codimension-zero bulk region. – 3 – ig. 1:
For AdS , the RT formula satisfies field-theory causality marginally. The plane generatedby null geodesics (color-coded by angular momentum) from a given boundary point (blue)is also ruled by spacelike geodesics at constant time (color-coded by time). boundary causality. However, since entanglement entropy is a more nonlocal quantity,which according to HRT is captured by a bulk surface that can go behind event andapparent horizons [15, 16] and penetrate into causally disconnected regions from theboundary, it is far less obvious whether CFT causality will survive in this context.Let us first consider a static example. Although it is guaranteed to be consistentwith CFT causality since it is covered by the RT prescription which is “derived” fromfirst principles, it is useful to gain appreciation for how innocuous or far-fetched causal-ity violation would appear in the more general case. Intriguingly, already the simplestcase of pure AdS reveals the potential for things to go wrong. As illustrated in Fig.1, the null congruence from a single boundary point (which bounds the bulk regionwhich a boundary source at that point can influence) is simultaneously foliated byspacelike geodesics {E A } . So a signal that can influence a given extremal surface E A inthat set can also influence ∂ A , thereby upholding CFT causality. However, note thathere causality was maintained marginally: if the extremal surface was deformed awayfrom A by arbitrarily small amount, one would immediately be in danger of causalityviolation.Another, less trivial, test case is the static eternal Schwarzschild-AdS black hole.The extremal surface that encodes entanglement between the two boundaries is thehorizon bifurcation surface. Again, arbitrarily small deformation of this surface wouldshift it into causal contact with at least one of the boundaries, thereby endangeringcausality; in particular, entanglement entropy for one CFT should not be influencedby deformations in the other CFT. For static geometries we’re in fact safe because– 4 –xtremal surfaces do not penetrate event horizons [17]; however this is no longer thecase in dynamical situations [15, 16, 18–20]. Moreover, as illustrated in [21], in Vaidya-AdS geometry, E A can be null-related to the past tip of D [ A ], thereby again upholdingcausality just marginally—an arbitrarily small outward deformation of the extremalsurface would render it causally accessible from D [ A ]. These considerations demon-strate that the question of whether the HRT prescription is consistent with field-theorycausality is a highly nontrivial one.The main result of this paper is a proof that, if the bulk spacetime metric obeysthe null energy condition, then the extremal surface E A does indeed obey both of theabove requirements. We conclude that the HRT formula is consistent with field-theorycausality. This theorem can be viewed as a generalization of the Gao-Wald theorem[14]. We regard it as a highly nontrivial piece of evidence in favor of the HRT formula.Along the way, we will also slightly sharpen the statement of the HRT formula, and inparticular clarify the homology condition on E A .Partial progress towards this result was achieved in [22, 23], which showed thatthe extremal surface E A generically lies outside of the “causal wedge” of D [ A ], theintersection of the bulk causal future and causal past of D [ A ]. (However, these worksdid not make the connection to field-theory causality). A stronger statement equivalentto our theorem was proved in [12] (cf., Theorem 6) and it is noted in passing that thiswould ensure field theory causality. We present an alternate proof which brings outsome of of the bulk regions more cleanly and make the connections with boundarycausality more manifest.As a byproduct of our analysis, we will identify a certain bulk spacetime region,which we call the entanglement wedge and denote W E [ A ], which is bounded on one sideby D [ A ] and on the other by E A . Apart from providing a useful quantity in formulatingand deriving our results, the entanglement wedge is, as we will argue, the bulk regionmost naturally associated with the boundary reduced density matrix ρ A .The outline of this paper is as follows. We begin in § causalshadow , and showing that the HRT surface lies in this causal shadow. In §
3, webegin to develop some intuition used in the proof of our main theorem, by consideringclasses of null geodesic congruences in AdS . In § § ote added: While this paper was nearing completion [24] appeared on the arXiv,which has some overlap with the present work. It introduces the notion of quantumextremal surfaces and argues that for bulk theories that satisfy the generalized secondlaw such surfaces satisfy the causality constraint.
In this section we will state our basic results and discuss some of their implications.The specific proof, and some additional results, will be presented in §
4. In § § § § § Consider a local quantum field theory (QFT) on a d -dimensional globally hyperbolicspacetime B . The state on a given Cauchy slice Σ is described by a density matrix ρ Σ ;this could be a pure or mixed state. We are interested in the entanglement between thedegrees of freedom in a region A ⊂
Σ and its complement A c . Following establishedterminology, we call the boundary ∂ A the entangling surface .The entanglement entropy is defined by first decomposing the Hilbert space H ofthe QFT into H A ⊗ H A c , after imposing some suitable cutoff. The reduced densitymatrix ρ A := Tr H A c ρ Σ captures the entanglement between A and A c ; in particular, theentanglement entropy is given by its von Neumann entropy: S A := − Tr ( ρ A ln ρ A ). For Throughout this paper we will require all Cauchy slices to be acausal (no two points are connectedby a causal curve). This is slightly different from the standard definition in the general-relativityliterature, in which a Cauchy slice is merely required to be achronal. The reason is to ensure thatdifferent points represent independent degrees of freedom, which is useful when we decompose theHilbert space according to subsets of the Cauchy slice. Technically, A is defined as the interior of a codimension-zero submanifold-with-boundary in Σ, ∂ A is the boundary of that submanifold, and A c := Σ \ ( A ∪ ∂ A ). In the case of gauge fields, this decomposition is not possible even on the lattice. Instead, one mustextend the Hilbert spaces H A , H A c to each include degrees of freedom on ∂ A , so that H ⊂ H A ⊗ H A c [26–29]. – 6 –olographic theories, we expect that this quantity has good properties in the large- N limit, unlike the R´enyi entropies S n, A := − n − ln Tr ( ρ n A ) [10, 31]. Note that bothquantities are determined by the eigenvalues of ρ A , and are thus insensitive to unitarytransformations of ρ A .Now, since Σ is a Cauchy slice, the future (past) evolution of initial data on it allowsus to reconstruct the state of the QFT on the entirety of B . In other words, the pastand future domains of dependence of Σ , D ± [Σ], together make up the backgroundspacetime on which the QFT lives, i.e., D + [Σ] ∪ D − [Σ] = B . Likewise, the domainof dependence of A , D [ A ] = D + [ A ] ∪ D − [ A ], is the region where the reduced densitymatrix ρ A can be uniquely evolved once we know the Hamiltonian acting on the reducedsystem in A . A c similarly has its domain of dependence D [ A c ]. However, unless A comprises theentire Cauchy slice, the two domains do not make up the full spacetime, D [ A ] ∪ D [ A c ] (cid:54) = B , since we have to account for the regions which can be influenced by the entanglingsurface ∂ A . Denoting the causal future (past) of a point p ∈ B by J ± ( p ) we find thatwe have to keep track of the regions J ± [ ∂ A ] which are not contained in either D [ A ] or D [ A c ]. As a result, the full spacetime B decomposes into four causally-defined regions:the domains of dependence of the region and its complement, and the causal futureand past of the entangling surface: B = D [ A ] ∪ D [ A c ] ∪ J + [ ∂ A ] ∪ J − [ ∂ A ] . (2.1)These four regions are non-overlapping (except that J ± [ ∂ A ] both include ∂ A ). SeeFig. 2 for an illustration of this decomposition. Although this decomposition is fairlyobvious pictorially, for completeness we provide a proof in § ρ A , and hencethe R´enyi and von Neumann entropies, are invariant under unitary transformationswhich act on H A alone or on H A c alone. These include perturbations of the Hamilto-nian and local unitary transformations supported in the domains D [ A ] or D [ A c ]. Inparticular, if we consider another region A (cid:48) of a Cauchy slice Σ (cid:48) such that D [ A ] = D [ A c ](as indicated in Fig. 2), then the state ρ Σ (cid:48) is related by a unitary transformation tothe state ρ Σ . It is clear that such a transformation can be constructed from operatorslocalized in A , and so does not change the entanglement spectrum of ρ A . Furthermore, Technically, by “large- N ” we mean large c eff , where c eff is a general count of the degrees of freedom(see [30] for the general definition of c eff ). We remind the reader that D [ A ] is defined as the set of points in B through which every inex-tendible causal curve intersects A . Note that, given that we have defined A as an open subset of Σ, D [ A ] is open subset of B . – 7 – + [ ∂ A ] J − [ ∂ A ] Fig. 2:
An illustration of the causal domains associated with a region A , making manifest thedecomposition of the spacetime into the four distinct domains indicated in (2.1) . Twodeformations A (cid:48) are also included for illustration in the right panel. if we fix the state at t → −∞ , then a perturbation to the Hamiltonian with support R cannot affect the state on a Cauchy slice to the past of R (i.e. that doesn’t intersect J + [ R ]). Such a perturbation can therefore affect the entanglement spectrum only if R intersects J − [ ∂ A ], because otherwise we can imagine evaluating S A by using a suffi-ciently early Cauchy slice Σ (cid:48) ⊃ ∂ A that passes to the past of R . Similarly, if we fix thestate at t → + ∞ , the spectrum can be affected only by perturbations in J + [ ∂ A ]. Insummary, we have the following properties of ρ A : • The entanglement spectrum of ρ A depends only on the domain D [ A ] and not onthe particular choice of Cauchy slice Σ. The spectrum is thus a so-called “wedgeobservable” (although it is not, of course, an observable in the usual sense). • Fixing the state in either the far past or the far future, the entanglement spectrumof ρ A is insensitive to any local deformations of the Hamiltonian in D [ A ] or D [ A c ].These are the crucial causality requirements that entanglement (R´enyi) entropies arerequired to satisfy in any relativistic QFT.The essential result of this paper is that the HRT proposal for computing S A satisfies these causality constraints. In the conclusions we will revisit the question ofwhat the dual of ρ A , and thus of the data in D [ A ], might be. Let us now restrict attention to the class of holographic QFTs, which are theoriesdual to classical dynamics in some bulk asymptotically AdS spacetime. To be precise,we only consider strongly coupled QFTs in which the classical gravitational dynamics– 8 –runcates to that of Einstein gravity, possibly coupled to matter which we will assumesatisfies the null energy condition.The dynamics of the QFT on B is described by classical gravitational dynamicson a bulk asymptotically locally AdS spacetime M with conformal boundary B , thespacetime where the field theory lives. We define ˜ M := M ∪ B . ˜ M is endowed with ametric ˜ g ab which is related by a Weyl transformation to the physical metric g ab on M ,˜ g ab = Ω g ab , where Ω → B . Causal domains on ˜ M will be denoted with a tilde todistinguish them from their boundary counterparts, e.g., ˜ J ± ( p ) will denote the causalfuture and past of a point p in ˜ M and ˜ D [ R ] will denote the domain of dependence ofsome set R ⊂ ˜ M .It will also be useful to introduce a compact notation to indicate when two points p and q are spacelike-separated; for this we adopt the notation (cid:16) , i.e. p (cid:16) q ⇔ (cid:64) a causal curve between p and q . (2.2)Moreover, to denote regions that are spacelike separated from a point, we will use S ( p )and ˜ S ( p ) in the boundary and bulk respectively, S ( p ) := { q | p (cid:16) q } = (cid:0) J + ( p ) ∪ J − ( p ) (cid:1) c and ˜ S ( p ) := (cid:16) ˜ J + ( p ) ∪ ˜ J − ( p ) (cid:17) c . (2.3)Just as for other causal sets, we can extend these definitions to any region R , namely S [ R ] := ∩ p ∈ R S ( p ) is the set of points which are causally disconnected from the entireregion R , etc.Having established our notation for general causal relations, let us now specify thenotation relevant for holographic entanglement entropy. As before we will fix a region A on the boundary. The HRT proposal [3] states that the entanglement entropy S A isholographically computed by the area of a bulk codimension-two extremal surface E A that is anchored on ∂ A ; specifically, S A = Area( E A )4 G N . (2.4)In the static (RT) case, it is known that the extremal surface is required to be homol-ogous to A , meaning that there exists a bulk region R A such that ∂ R A = A ∪ E A .So far, it has not been entirely clear what the correct covariant generalization of thiscondition is. In particular, should it merely be a topological condition, or should oneimpose geometrical or causal requirements on R A , for example, that it be spacelike?(A critical discussion of the issues involved can be found in [32].) In this paper, we These are necessary but not sufficient conditions for the spacetime to be asymptotically AdS. – 9 –ill show that a clean picture, consistent with all aspects of field-theory causality, isobtained by requiring that R A be a region of a bulk Cauchy slice. We will call thisthe “spacelike homology” condition. The homology surface R A naturally leads us to the key construct pertaining toentanglement entropy, which we call the entanglement wedge of A , denoted by W E [ A ].This can be defined as a causal set, namely the bulk domain of dependence of R A , W E [ A ] := ˜ D [ R A ] . (2.5)Note that the entanglement wedge is a bulk codimension-zero spacetime region, whichcan be equivalently identified with the region defined by the set of bulk points whichare spacelike-separated from E A and connected to D [ A ]. The latter definition has theadvantage of absolving us of having to specify an arbitrary homology surface R A ratherthan just E A and D [ A ]. As we shall see below, the bulk spacetime can be naturallydecomposed into four regions analogously to the boundary decomposition (2.1); theentanglement wedge is then the region associated with (and ending on) D [ A ].While we have focused on the regions in the bulk which enter the holographicentanglement entropy constructions, we pause here to note two other causal constructsthat can be naturally associated with A . First of all we have the causal wedge W C [ A ]which is set of all bulk points which can both send signals to and receive signals fromboundary points contained in D [ A ], i.e., W C [ A ] := ˜ J + (cid:2) D [ A ] (cid:3) ∩ ˜ J − (cid:2) D [ A ] (cid:3) . (2.6)(The entanglement wedge W E [ A ] and causal wedge W C [ A ] are in fact special cases of the“rim wedge” and “strip wedge” introduced recently in [33] as bulk regions associatedwith residual entropy.)The second bulk causal domain which will play a major role in our discussion belowis a region we call the causal shadow Q ∂ A associated with the entangling surface ∂ A . Technically, similarly to A , we define R A to be the interior of a codimension-zero submanifold-with-boundary of a Cauchy slice ˜Σ of ˜ M (with ˜Σ ∩ B = Σ). Since ˜Σ itself has a boundary (namely itsintersection with B ), the interior of a subset (in the sense of point-set topology) includes the part ofits boundary along B . Thus, R A includes A (but not E A ). If there are multiple extremal surfaces obeying the spacelike homology condition, then we are topick the one with smallest area. However, in this paper we will not use this additional minimalityrequirement; all our theorems apply to any spacelike-homologous extremal surface. While we have associated it notationally with the region A , it depends only on D [ A ]. Following [22], we can also define a particular bulk codimension-two surface Ξ A , the causalinformation surface, to be the rim of the causal wedge; in fact, it is the minimal area codimension-twosurface lying on ∂ W C [ A ]. – 10 – A AA c ← D [ A ] D [ A c ] → (cid:121) Q ∂ A Fig. 3:
Example of a causally trivial spacetime and a boundary region A whose causal shadow isa finite spacetime region. We have engineered an asymptotically AdS geometry sourcedby matter satisfying the null energy condition (see footnote 14) and taken A to nearlyhalf the boundary, ϕ A = 1 . , at t = 0 (thick red curve). The shaded regions on theboundary cylinder are D [ A ] and D [ A c ] respectively. The extremal surface is the thickblue curve, while the purple curves are the rims of the causal wedge (causal informationsurfaces) for A and A c respectively. A few representative generators are provided fororientation: the blue null geodesics generate the boundary of the causal wedge for A while the green ones do likewise for A c . The orange generators in the middle of thespacetime generate the boundary of the causal shadow region Q ∂ A . We define this region as the set of points in the bulk M that are spacelike-related toboth D [ A ] and D [ A c ], i.e., Q ∂ A := (cid:16) ˜ J + [ D [ A ]] ∪ ˜ J − [ D [ A ]] ∪ ˜ J + [ D [ A c ]] ∪ ˜ J − [ D [ A c ]] (cid:17) c = ˜ S [ D [ A ] ∪ D [ A c ]] . (2.7)For a generic region A in a generic asymptotically AdS spacetime, the causal shadow– 11 –s a codimension-zero spacetime region; see Fig. 3 for an illustrative example. Incertain special (but familiar) situations, such as spherically symmetric regions in pureAdS (where ρ A is unitarily equivalent to a thermal density matrix), it can degenerateto a codimension-two surface. In such special cases, the entanglement wedge and thecausal wedge coincide [22]. In general, the causal information surface for A and thatfor A c comprise the edges of the causal shadow. For a generic pure state these causalinformation surfaces each recede from E A towards their respective boundary region butapproach each other near the AdS boundary. Hence the geometrical structure of Q ∂ A ,described in language of a three-dimensional bulk, is a “tube” (connecting the twocomponents of ∂ A ) with a diamond cross-section, which shrinks to a point where thetube meets the AdS boundary at ∂ A .For topologically trivial deformations of AdS, in the absence of E A (i.e. when thestate is pure and A = Σ) the causal shadow disappears, but intriguingly, even when A is the entire boundary Cauchy slice, the causal shadow can be nontrivial. This occursfor example in the AdS -geon spacetimes [34] and in perturbations of the eternal AdSblack hole, such as those studied by [35]. In such a situation we simply define the casualshadow of the entire boundary (dropping the subscript) as Q := ˜ S [ B ] = (cid:16) ˜ J + [ B ] ∪ ˜ J − [ B ] (cid:17) c (2.8)Here B is understood generally to include multiple disconnected components; the causalshadow is the region spacelike separated from points on all the boundaries. Having developed the various causal concepts which we require, let us now ask whatthe constraints of field-theory causality concerning entanglement entropy translate toin the bulk. The first constraint is that S A should be a wedge observable, i.e. if D [ A ] = D [ A (cid:48) ] then S A = S A (cid:48) . For this to hold in general, we need E A = E A (cid:48) . Thesecond concerns perturbations of the field-theory Hamiltonian. Such perturbations willsource perturbations of the bulk fields, including the metric, that will travel causallywith respect to the background metric. In particular, disturbances originating in D [ A ] The bulk metric used in the plot for Fig. 3 is ds = 1cos ρ (cid:18) − f ( ρ ) dt + dρ f ( ρ ) + sin ρ dϕ (cid:19) , f ( ρ ) = 1 −
12 sin (2 ρ ) . The matter supporting this geometry satisfies the null energy condition as can be checked explicitly. Since these describe pure states, the presence of a causal shadow region does not necessarilyguarantee the presence of an extremal surface whose area gives the entanglement entropy containedwithin it. However, there will be some extremal surface spanning this region. – 12 –ill be dual to bulk modes propagating in ˜ J + (cid:2) D [ A ] (cid:3) (if we fix the state in the far past)or in ˜ J − (cid:2) D [ A ] (cid:3) (if we fix the state in the far future). If either of these bulk regionsintersected E A , the dual of local operator insertions in D [ A ] could change the area of E A , meaning that the HRT proposal would be inconsistent with causality in the QFT.By the same token, the extremal surface cannot intersect ˜ J + (cid:2) D [ A c ] (cid:3) or ˜ J − (cid:2) D [ A c ] (cid:3) .Since the region complement to union of the causal sets ˜ J ± [ D [ A ]] , ˜ J ± [ D [ A c ]] is the setof points that are spacelike related to D [ A ] ∪ D [ A c ], we learn that E A (cid:16) D [ A ] ∪ D [ A c ] . (2.9)In others words, using (2.7) we can say that E A has to lie in the causal shadow of ∂ AE A ⊂ Q ∂ A . (2.10)It is known, based on properties of extremal surfaces, that E A lies outside the causalwedges W C [ A ] and W C [ A c ] [12, 22, 23]. This leaves open the possibility that the surfacecould still lie in the causal future (or past) of the boundary domain of dependence of A or A c . A particular worry arises in explicit examples in Vaidya-AdS geometries wherethe extremal surface lies on the boundary of ˜ J + (cid:2) D [ A ] (cid:3) . This then leaves open thequestion whether one might indeed be able to push E A into a causally forbidden region,by introducing appropriate deformations in D [ A ]. A theorem of Wall [12] (Theorem 6of the reference), guarantees that this does not occur (modulo some assumptions).We will prove an essentially equivalent statement in §
4, directly for extremal sur-faces in an asymptotically AdS spacetime. The main result however can be stated interms of three simple causal relations:˜ D [ R A ] ∩ B = D [ A ]˜ D [ R c A ] ∩ B = D [ A c ]˜ J ± [ E A ] ∩ B = J ± [ ∂ A ] . (2.11)In other words, the causal split of the bulk into spacelike- and timelike-separated regionsfrom E A restricts to the boundary at precisely the boundary split (2.1). Given thedecomposition (2.1), these causal relations imply that perturbations in D [ A ] ∪ D [ A c ]are not in causal contact with E A . So, as required, the extremal surface lies in thecausal shadow.As a consequence of this theorem, we will also show that, if there is a spacelikeregion A (cid:48) such that D [ A (cid:48) ] = D [ A ], then there is a bulk region R A (cid:48) such that ∂ R A (cid:48) = A (cid:48) ∪ E A , so E A is spacelike-homologous to A (cid:48) . Thus, the HRT formula gives the sameentanglement entropy for A (cid:48) and A , as required on the field-theory side.– 13 – .4 Entanglement for disconnected boundary regions A striking consequence of the theorems discussed above emerges when we considerspacetimes with two boundary components, and let A be (a Cauchy slice for) all of onecomponent.As a starting point, consider the eternal Schwarzschild-AdS d +1 black hole in theHartle-Hawking state, with a Penrose diagram shown in Fig. 4(a) below. The left andright boundaries of the diagram each have the topology S d − × R . This geometry isbelieved to be dual to the CFT on the product spatial geometry S d − L × S d − R , in theentangled “thermofield double” state [36–39]: | HH (cid:105) L,R = (cid:88) i e − β E i | E i (cid:105) L | E i (cid:105) R (2.12)where | E i (cid:105) R,L is the energy eigenstate of the CFT on S d − R,L .Let Σ R lie on the t = 0 slice of the right boundary, and consider the reduced densitymatrix for some region A ⊂ Σ R . Since this is a static geometry, its entanglemententropy S A is computed by a minimal surface E A which never penetrates past thebifurcation surface X of the black hole [17]. If we let A be the full Cauchy slice ofone of the boundaries, say A = Σ R , the extremal surface precisely coincides with theblack hole bifurcation surface, as indicated in Fig. 4. Note that E A lies on the edge ofthe causally acceptable region since X sits at the boundary of both W C [ A ] and W C [ A c ],and therefore constitutes the entire causal shadow for this special case.One might now wonder what happens if we deform the state (2.12). This is notan innocuous question. In time-dependent geometries, the global (teleological) natureof the event horizon implies that extremal surfaces anchored on the boundary can passthrough this horizon [15]. Furthermore, as first explicitly shown in [16], even apparenthorizons do not form a barrier to the extremal surfaces. Hence we see that, a priori, ina state which is a deformation of (2.12), E A is in danger of entering W C [ A c ].The theorems we have stated above indicate that this does not happen. Thequestion is, how precisely does the extremal surface E A avoid doing so? As a first stepto answering this, consider a deformation of the static eternal case localized along a nullshell emitted from the right boundary at some time. The corresponding metric is givenby the global Vaidya-SAdS geometry, where both the initial (prior to the shell) andfinal (after the shell) spacetime regions describe a black hole. Fig. 4b presents a sketchof the Penrose diagram of such a geometry, contrasted with the standard static eternalSchwarzschild-AdS black hole (Fig. 4a). The diagonal brown line represents the shell Note that the extremal surface does not come arbitrarily close to the horizon—it either includesa component that wraps the horizon, or stays a finite distance away from it [32]. – 14 –a) (b)
FP RL F b F a F c PP c R a R c R b L Fig. 4:
Sketch of Penrose diagram for (a) static eternal Schwarzschild-AdS and (b) ‘thinshell’ Vaidya-Schwarzschild-AdS, with the various regions labeled. The AdS boundariesare represented by vertical black lines, the singularities by purple curves, the horizons bydiagonal blue lines, and the ‘shell’ in the Vaidya case by diagonal brown line. which is sourced at some time on the right boundary and implodes into the black hole(terminating at the future singularity), and the blue lines represent the various (futureand past, left and right) event horizons. The solid parts of these lines indicate wherethese event horizons coincide with apparent horizons (as well as isolated horizons); thedashed parts are parts of the event horizon which are not apparent horizons.In such a geometry, let us again consider A = Σ R . Then our theorems guaranteethat the extremal surface must lie on the null sheet separating regions R c and P c : itis again spacelike-separated from both D [Σ L ] and D [Σ R ]. (In fact, since the spacetimeprior to the shell is identical to the eternal static case, the extremal surface remains inthe same location as for the static case, namely the bifurcation surface where regions R c and L touch.) The situation is again marginal, much like the original undeformedcase. Indeed, any perturbation to Schwarzschild-AdS which emanates from (or reachesto) the right boundary cannot change the location of the original extremal surface bycausality; it could at most generate a new extremal surface.A less marginal case occurs when we symmetrically perturb both copies of theCFT as above. Consider a perturbation at t = 0 such that spherically symmetric nullshells are emitted both to the past and future on both sides of the diagram. One thenobtains the Penrose diagram shown in Fig. 5; this has time-reflection symmetry about– 15 –FT R CFT L AE A Q W C [ A ] W C [ A c ] F A Fig. 5:
Sketch of Penrose diagram for a symmetric Vaidya-Schwarzschild-AdS geometry ob-tained by imploding null shells to the past and future from both boundaries. The crucialnew feature of note is the presence a causal shadow region that is spacelike separatedfrom both boundaries. We have also indicated the extremal surface E A for the region A = Σ R in red at the center of the figure and F A is a S d − of finite area in the causalfuture of the left boundary. The lightly shaded regions are the causal wedges associatedwith A and A c respectively. t = 0, symmetry under exchanging the left and right sides, and the SO ( d ) rotationalsymmetry.According to the theorems above, the extremal surface must be spacelike-separatedfrom both boundaries, when we take A = Σ R . Using both time and space reflectionsymmetry, it is clear that E A must sit in the center of the causal shadow Q of the twoboundaries, spacelike separated from both.In the general case of spherically symmetric spacetime (even in the absence of timeor space reflection symmetry) there is an easy proof of our claim that E A must lie in thecausal shadow. We proceed by contradiction: suppose that a spherical extremal surface E A lies in ˜ J + [Σ L ]. This means that on a Penrose diagram, it lies somewhere in the top-left region; say it is the surface F A indicated in Fig. 5 (which by rotational symmetry isa copy of S d − ). Let us then consider the past congruence of null normal geodesics from F A towards B L . Since we assume that F A candidate surface lies in ˜ J + [Σ L ], past-goingnull congruences from the surface intersect B L on a spacelike codimension-one surface.– 16 –n other words, the area of the spheres grows without bound along this past-directedcongruence.However, by definition, for an extremal surface the initial expansion is vanishing.Moreover, if the matter in the spacetime satisfies the null energy condition,then it alsofollows that the area along the congruence is guaranteed not to grow. Nor can the areago to zero along the congruence, since the area of the S d − represented by each pointon the Penrose diagram is finite. It therefore follows that our assumption about E A penetrating ˜ J + [Σ L ] must be erroneous; F A cannot be an extremal surface. Running asimilar argument for the other unshaded regions in Fig. 5, we learn that the extremalsurface must indeed lie in the causal shadow region, as denoted by the red surface E A .Indeed, in this particular case, the extremal surface lies at the point on the Penrosediagram where the future and past apparent horizons meet—the “apparent bifurcationsurface”. The fact that it lies in the causal shadow is a consequence of the familiarfact that the apparent horizon can never be outside the event horizon, applied to bothfuture and past horizons.While the above result relied on the special properties of spherically symmetry(both of the spacetime and the null congruences therein), the theorems we prove in § § § ,in order to develop a picture of the relevant causal domains, before embarking on ageneral proof in § In this section, we consider null geodesic congruences emanating from curves in AdS that are anchored at the boundary. Our aim is to build some intuition about suchcongruences in a simple setting, since their properties will play a crucial role in theproofs in what follows. Readers familiar with the general statements are invited to skipahead to the abstract discussion.We work in the Poincar´e patch of AdS with the standard metric: ds = 1 z (cid:0) − dt + dx + dz (cid:1) (3.1)Since our aim is to understand specifically the (causal) boundary of bulk causal do-mains, we are going to examine properties of null geodesic congruences. In particular,– 17 –or a spacelike codimension-one region R ⊂ M which is anchored on the AdS boundary,the domain of dependence ˜ D [ R ] is bounded by a family of outgoing null geodesics ema-nating from ∂R , up to the point where each geodesic encounters a caustic or intersectsanother generator. To gain intuition for how these null congruences behave in the context of theextremal surfaces of interest, we examine a more general family of codimension-twosurfaces (these are curves in AdS ) which in the above coordinates are given by x + z a = 1 , t = 0 (3.2)parameterized by a . Note that all of these are anchored on the boundary R , at theends of the interval A = { ( t, x ) ∈ R , | t = 0 , x ∈ [ − , } . (For orientation, seethe bottom set of curves in Fig. 7.) When a = 1, the surface is a semi-circle, whichis simultaneously the causal information surface Ξ A defined in [22], and the extremalsurface E A for the region A under consideration. Surfaces with a < W C [ A ], while those with a > D [ A ]. We wish to study the family of null congruences leaving these surfaces, as wevary a . The geodesics will be labelled by their starting position x and parameterizedby an affine parameter λ (fixed such that we have unit energy along each geodesic). Since the a = 1 surface is extremal, the null expansion Θ( λ ; a = 1) = 0 for eachgenerator. For the surfaces with a <
1, closer to the boundary, we expect that theexpansion is positive and the congruence intersects the boundary in a spacelike curveinside D [ A ] = { ( t, x ) ∈ R , | | t ± x | ≤ } . For curves with a >
1, long ellipse, weexpect the expansion to be negative. The resulting congruence should develop a causticbefore reaching the boundary.Due to the relative simplicity of the set-up, we can confirm these expectationsexplicitly. Since everything is time-symmetric, let us consider just the future-directed The latter set of intersections is referred to as cross-over points; the set of these generically forma crossover seam which is codimension-one on this null surface. – 18 –utgoing congruence: z ( λ ) = a (cid:112) − x (cid:112) − x + a x a (1 − x ) λ + (cid:112) − x + a x x ( λ ) = x a (1 − a ) (1 − x ) λ + (cid:112) − x + a x a (1 − x ) λ + (cid:112) − x + a x t ( λ ) = a (1 − x ) (cid:112) − x + a x λa (1 − x ) λ + (cid:112) − x + a x (3.3)Note that the endpoints of these generators at λ = ∞ are given by z ∞ = 0 , x ∞ = x (1 − a ) , t ∞ = a (cid:113) − x + a x (3.4)A representative plot of the generators is given in Fig. 6 for a = 0 . a = 1 . a <
1, the generators don’t intersect each other beforereaching the boundary, and they reach within D + [ A ]. On the other hand, when a > cross-overpoints ; non-neighbouring geodesics intersect at these points. This seam terminates ina caustic , which as always refers to the locus where neighbouring geodesics intersect. We can determine the intersection between distinct geodesics in the bulk using theexplicit expressions from (3.3). By symmetry of the set-up, we know that geodesicswith opposite values of x necessarily intersect, and they must do so at x = x × = 0.Solving for the intersection of the pair of geodesics starting from x and − x we findthat they meet at: t × = (cid:112) − x + a x a , z × = a − a (cid:113) − x , λ × = (cid:112) − x + a x a ( a −
1) (1 − x ) (3.5)This generates the seam of cross-over points depicted in the right panel of Fig. 6,and plotted for various values of a in Fig. 7 (the top set of curves, color-coded by a corresponding to the initial surface indicated by the thick horizontal curve of thesame color). It is easy to see from (3.5) that the cross-over points terminate on theboundary at the future tip of D + [ A ], i.e., at z = 0 , x = 0 , t = 1, corresponding tothe intersection of the boundary geodesics x = ±
1. On the other hand, the cross-over– 19 – ig. 6:
Null normal congruence from the initial surface given by (3.2) with a = 0 . (left) and a = 1 . (right). The initial surface is the bold black curve on the bottom, the boundaryis the shaded plane on the left in each plot (with the domain of dependence D + [ A ] boundary indicated by the thin black lines), the individual geodesics are the thin linescolor-coded by x , their endpoints on the boundary are depicted by the red curve, andfinally the seam of crossover points where generators intersect for a > is the blue thickcurve. (The generators are cut off at a finite value of λ ≈ , so in the plot they don’tlook like they reach all the way to the boundary.) seams for different a start at the point in the bulk when neighbouring geodesics from x (cid:39) x × = 0 , t × = 1 a , z × = a − a , λ × = 1 a ( a −
1) (3.6)To summarize, depending on whether a is greater or less than 1, the congruence hasqualitatively different behaviour, as illustrated in Fig. 7. For a < D + [ A ], while for a >
1, the generators intersect each other at the seam of crossover points (depicted bycolors from red toward purple). At precisely a = 1, all generators reach the boundaryat the future tip of D + [ A ], namely z = 0 , x = 0 , t = 1.– 20 – ig. 7: Initial surfaces (thick curves at the bottom, color-coded by a ), along with endpointsof the generators of the corresponding null congruence: for a = 1 (initial surface is thered semi-circle), all generators meet at the tip. Increasing a > (color shift towardspurple and blue) makes the generators intersect at the seam of cross-over points beforereaching the boundary. On the other hand, decreasing a < (color shift towards orangeand green) makes the generators reach the boundary within D + [ A ] (depicted as inFig. 6). Let us now analyze the expansion along this congruence. This can be calculated as thechange in area along the wavefrontΘ( λ, x ) = 1 A ( λ, x ) ∂∂λ A ( λ, x ) (3.7)with A ( λ, x ) = (cid:90) x + δxx (cid:115) − t (cid:48) ( λ, ˜ x ) + x (cid:48) ( λ, ˜ x ) + z (cid:48) ( λ, ˜ x ) z ( λ, ˜ x ) d ˜ x (3.8)where t (cid:48) ( λ, x ) ≡ ∂∂x t ( λ ; x ) etc., using the expressions given in (3.3). While one cannumerically solve for Θ( λ ) it is easier to obtain the solution for small λ and evolveusing the Raychaudhuri equation.Near λ = 0, the leading order expression for Θ is:Θ ≡ Θ( λ = 0) = a (1 − a ) (1 − x ) (1 − x + a x ) / (3.9)This is plotted in the left panel of Fig. 8 (with same color-coding by a as employed inFig. 7). At the ends of the interval x = ±
1, Θ vanishes (which is to be expected since– 21 – (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:81) (cid:72) Λ(cid:61) (cid:76) Λ(cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:81) (cid:72) x (cid:61) (cid:76) Fig. 8:
Expansion Θ( λ ; x ) along the generators for various values of a (color-coded by a asin Fig. 7). On left, we show the expansion from the initial surface λ = 0 as a functionof the starting position x . On right, we fix x = 0 as plot the evolution of Θ( λ ) alongthe radial generator. the congruence approximates a larger one with a = 1), while Θ reaches its extremumat the midpoint, x = 0 (again, expected by symmetry), where Θ ( x = 0) = a (1 − a ).Furthermore, Θ is positive for a < a >
1; that is, the congruencesare expanding for a < a > as our initial condition, it is straightforward to solve the Raychaudhuriequation d Θ dλ = − Θ − σ ab σ ab − R ab ξ a ξ b (3.10)to find the expansion along the geodesics. Here ξ a is the tangent vector to the nullgeodesics and σ µν is the shear of the congruence. For a one-dimensional congruencethe shear trivially vanishes and the Ricci tensor contracted with null tangents likewisevanishes upon using the bulk equations of motion R ab = − g ab , so (3.10) simplifies to: d Θ dλ = − Θ ⇒ Θ( λ ) = Θ λ (3.11)Using (3.9), we find:Θ( λ, x ) = a (1 − a ) (1 − x ) (1 − x + a x ) / + a (1 − a ) (1 − x ) λ (3.12)In Fig. 8 we have plotted this as a function of λ for x = 0, at which Θ = a (1 − a )1+ a (1 − a ) λ .For a >
1, we expect the congruence to develop a caustic where the expansiondiverges. This occurs when infinitesimally nearby geodesics intersect each other. Eq.– 22 –3.12) shows that this can only occur for a >
1, where the second term in the de-nominator is negative for positive λ . In this case Θ( λ ) → −∞ at a finite value of λ = λ c , λ c = (1 − x + a x ) / a ( a −
1) (1 − x ) (3.13)for any x . The spacetime coordinates for the points along the congruence where thishappens are given by x c = (1 − a ) x , t c = (1 − x + a x ) / a , z c = a − a (1 − x ) / (3.14)Viewed as a pair of parametric curves parametrized by x which starts at x = 0 andends at x = ±
1, the caustic seams are null curves, starting at the intersection point(3.6) and ending on the boundary at z c = 0, x c = ± (1 − a ), and t c = a . Note thatthis is a finite distance on the boundary.The divergence Θ → −∞ signifies the presence of conjugate points, but their geo-metric meaning is a bit obscure in our discussion so far. The reason is as follows: aswe see in Fig. 6 and can check explicitly, we generically have caustics in the neighbour-hood of x (cid:39)
0, but more generally encounter cross-over points from the intersectiongeodesics symmetrically placed about x = 0. The expansion is finite along the cross-over seam (3.5) for x (cid:54) = 0. This can be understood by realizing that the expansion isa local property of the nearby geodesics which doesn’t know about any other piece ofthe congruence. So nothing special ought to happen at the cross-over points which arenon-local in the congruence, and indeed these are not conjugate points.The clue as to the geometric meaning of Θ → −∞ comes from plotting this locuson the surface of the null congruence (continued through the cross-over seam). Thisis presented in Fig. 9 by the thick red curves. We see that the surface intersects itselfat the cross-over seam, beyond which the constant- λ wavefronts form closed loops. Onthe sharp flank, these wavefronts turn around and locally become null; this is preciselywhere A ( λ, x ) vanishes and therefore Θ → −∞ . The upshot of our calculations can be summarized as follows. Consider the null geodesiccongruence emanating from a codimension-two spacelike surfaces F A ⊂ M anchoredon the boundary of a region A with ∂ A = F A ∩ B . • If F A ⊂ W C [ A ] then the congruence terminates inside D [ A ] along a spacelikeboundary codimension-one surface.– 23 – ig. 9: Surface generated by the null normal congruence, along with the locus of points onthis surface where the expansion diverges, indicated by the thick red curves. The cyancontours represent the geodesic generators, while the blue contours are the constant- λ wavefronts (we cut off the surface at | x | < for convenience). • If F A lies on the boundary of the causal wedge W C [ A ] then the congruence inter-sects the boundary on the null surface ∂D [ A ]. • If F A ⊂ ˜ S [ D [ A ]] then the congruence finds itself terminated by a seam of cross-over points (and if continued further, would encounter caustic points prior toreaching the AdS boundary). The seam itself however reaches out to the boundaryand ends on the future tip of ∂D [ A ].This gives a clear picture of the causal domains for regions bounded by curves insideand outside of W C [ A ]. As we will see in our explicit proof, the extremal surface willin general lie outside of W C [ A ]; in special cases it can at best lie on the boundary, butnever in the interior, of the causal wedge. We now get to the main part of the paper where we prove that the extremal surface E A satisfies the causality requirements discussed in § In higher-dimensional setting, D [ A ] itself may terminate in a crossover seam rather than a singlepoint, which occurs when the null generators of ∂D [ A ] on the boundary themselves cross over. – 24 –he causal relations quoted there in (2.11). These will establish for us the consistencyof the HRT proposal for computing holographic entanglement entropy.In § § § A implies a natural decomposition of the spacetime into four regions D [ A ], D [ A c ], and J ± [ ∂ A ]. Then, given the spacelike homology condition, and using the results of § In this subsection we will describe our holographic setup and assumptions. Let ( M , g ab ) be a connected spacetime, of dimension greater than or equal to 3,that can be embedded in a spacetime ( ¯ M , ˜ g ab ), such that the boundary B of M in¯ M is a smooth timelike hypersurface in ¯ M , and such that ˜ g ab = Ω g ab , where Ω is asmooth function on ¯ M that vanishes on B . (We do not assume that B is connected.)We define ˜ M := M ∪ B . On ˜ M we have a causal structure induced from ˜ g ab , which in M agrees with that induced from g ab . We make the following assumptions:(i) ( M , g ab ) obeys the null energy condition.(ii) ˜ M is globally hyperbolic.(iii) Every null geodesic in ( B , ˜ g ab ) is a geodesic in ( ˜ M , ˜ g ab ). We largely follow the setup and assumptions of section 3 of [14], with two exceptions: we removethe null generic condition and we add the condition that the boundary is totally geodesic for nullgeodesics (assumption (iii) below). Assumption (iii) is equivalent to the following property of the extrinsic curvature K ab of B in ˜ M :for any point p ∈ B and any null vector k a in the tangent space to B at p , K ab k a k b = 0. That it holdsfor an asymptotically AdS spacetime can be seen by working in Fefferman-Graham coordinates. If weset Ω = 1 /z , where z is the standard radial coordinate, then K ab = 0 (so all geodesics in B are geodesicsin ˜ M , i.e. B is totally geodesic). The property K ab = 0 is not preserved by Weyl transformations, and – 25 –e begin by showing that B is globally hyperbolic. We omit the proofs, which arevery simple, cf., [40]. (For brevity, we will only indicate one time direction for eachstatement below, but the time-reversed statements are clearly equally valid.) Lemma 1
For any set Υ ⊂ ˜ M , ˜ D + [Υ] ∩ B ⊂ D [Υ ∩ B ] . Lemma 2 If ˜Σ ⊂ ˜ M is closed and acausal, then ˜Σ ∩ B is closed and acausal in B . Corollary 3 If ˜Σ is a Cauchy slice for ˜ M , then ˜Σ ∩ B is a Cauchy slice for B . Corollary 4 B is globally hyperbolic. In this subsection, we will study null geodesics in ˜ M . Assumption (iii) has the followinguseful implication: Lemma 5
Any null geodesic in ˜ M either (1) lies entirely in B , or (2) does not intersect B except possibly at its endpoints, where it is not tangent to B .Proof: Given a point p in B and a non-zero null vector in the tangent space to B at p , there exists a null geodesic in B passing through p with that tangent vector. Byassumption (iii), it is a geodesic in ˜ M , and by the uniqueness of geodesics it is theonly one. Therefore no null geodesic passing through M can intersect B tangentially.Finally, since B is the boundary of ˜ M and is smooth, any smooth curve that intersects B at some point without ending there must be tangent to it. (cid:3) Now we constrain the behavior of congruences of null geodesics that pass through M , using the fact that the metric g ab obeys the null energy condition and the fact thata geodesic that reaches B travels an infinite affine parameter. Lemma 6
Consider a codimension-one congrence of future-directed null geodesics in ˜ M , each of which lies entirely in M except possibly at its endpoints. Suppose thatthe part of the congruence in M has the following properties: (1) its expansion withrespect to the metric g ab is nowhere positive; (2) at each point, every deviation vectoris spacelike and orthogonal to the tangent vector. Then the congruence intersects B ona set of isolated points. so does not hold for a general choice of Ω, but the weaker condition K ab k a k b = 0 does (as can be seeneither from a direct calculation or from the fact that the set of null geodesics is invariant under Weyltransformations). We remind the reader that, as explained in footnote 4, throughout this paper we require allCauchy slices to be acausal, not just achronal. – 26 – roof:
We begin by working in the metric g ab . Since the deviation vectors are ev-erywhere spacelike, the expansion Θ is finite everywhere. On any geodesic that reaches B , the affine parameter goes to infinity, so, by the null energy condition, Θ is nowherenegative, and therefore vanishes everywhere. Again using the null energy condition,the shear therefore vanishes everywhere also. Therefore, for any one-parameter familyof geodesics that reach B , the norm of the deviation vector X a is a positive constantalong each geodesic.We now return to ˜ M , and switch to the metric ˜ g ab . On B , X a has vanishing norm;being also orthogonal to the geodesic’s tangent vector T a , it is proportional to T a (since orthogonal null vectors are proportional). Without loss of generality, we choosethe affine parameter λ on each geodesic so that it intersects B at λ = 0; hence, at λ = 0, X a is tangent to B . However, by lemma 5, T a is not tangent to B . So X a = 0. Sincethis holds for every one-parameter family of geodesics, every connected set of geodesicsthat reach B intersects it at a point. (cid:3) As a warm-up for our main theorem of this subsection, we will now use lemma6 to prove a version of the Gao-Wald theorem [14] and a version of the topologicalcensorship theorem [41].
Theorem 7
For any point p ∈ B , ˜ J + ( p ) ∩ B = J + ( p ) .Proof: Clearly J + ( p ) ⊂ ˜ J + ( p ) ∩ B . Let t be a global time function on ˜ M . Thenif t ( q ) < t ( p ) we have q / ∈ ˜ J + ( p ). Therefore, each connected component of B containssome points not in ˜ J + ( p ). Therefore, if ˜ J + ( p ) ∩ B (cid:54) = J + ( p ), then ∂ ˜ J + ( p ) ∩ B includesa hypersurface S in B that is not in J + ( p ). We will now show that S cannot exist. ∂ ˜ J + ( p ) consists of future-directed null geodesics starting at p on which, except atthe endpoints, every deviation vector is spacelike and orthogonal to the tangent vector.By lemma 5, each such geodesic either lies entirely in B or lies entirely in M except atits endpoints. In particular, the points in S must lie on geodesics that are entirely in M except at their endpoints. We thus consider the congruence of geodesics in M startingat p . Reversing its direction, every geodesic in this congruence reaches B (at p ), so theexpansion is nowhere negative. Therefore, in the forward direction, its expansion isnowhere positive. Thus the conditions of lemma 6 apply. Hence S consists of isolatedpoints, contradicting the fact that it is a hypersurface in B . (cid:3) Corollary 8 If B , B are distinct connected components of B , then ˜ J + ( B ) ∩ B = ∅ . Corollary 8 rules out traversable wormholes through the bulk connecting differentboundary components, and is thus closely related to topological censorship. (A simpleargument establishing this can be found in [42].)– 27 –ur goal for the rest of this subsection is generalize Theorem 7 to codimension-twosurfaces that are extremal with respect to g ab . First, we need two lemmas: Lemma 9
Let E be a compact codimension-two submanifold-with-boundary of ˜ M , withboundary N . Then every point p ∈ ∂ ˜ J + [ E ] is on a future-directed null geodesic lyingentirely in ∂ ˜ J + [ E ] that either (1) starts orthogonally from E and has no point conjugateto E between E and p , or (2) starts orthogonally from N , moving away from E (i.e. U a T a > , where T a is the tangent vector to the geodesic at its starting point, and U a is a vector at the same point that is tangent to E , normal to N , and outward-directedfrom E ).Proof: This is a generalization of theorem 9.3.11 in [25]. Every p ∈ ∂ ˜ J [ E ] lies ona null geodesic starting from E . If neither condition (1) nor (2) is met, then it can bedeformed to a timelike curve and therefore p ∈ ˜ I + [ E ]. (cid:3) Lemma 10
Let E be a spacelike submanifold-with-boundary of ˜ M whose restriction to M is extremal with respect to the metric g ab . Then E intersects B orthogonally, i.e.,every normal vector to E is tangent to B .Proof: A short calculation shows that, in M , the mean curvature ˜ K a of E withrespect to ˜ g ab is related to that with respect to g ab , K a , as follows:˜ K a = Ω − K a + dim( E ) ˜ Q ab ∂ b ln Ω , (4.1)where ˜ Q ab := Q ac ˜ g bc and Q ac is the projector normal to E . Since E is extremal, K a = 0.So ˜ K = dim( E ) ˜ Q ab ∂ a ln Ω ∂ b ln Ω . (4.2)Since E is smooth, ˜ K remains finite on B , where ln Ω → −∞ . This requires that everynormal vector to E be tangent to B . (cid:3) Theorem 11
Let E be a compact smooth spacelike codimension-two submanifold-with-boundary in ˜ M , whose only boundary is where it intersects B , and whose restriction to M is extremal with respect to the metric g ab . Then ˜ J + [ E ] ∩ B = J + [ E ∩ B ] .Proof: The proof is largely a repetition of that of Theorem 7. Clearly J + [ E ∩ B ] ⊂ ˜ J + [ E ] ∩ B . Let t be a global time function on ˜ M . Since E is compact, it has a minimumtime t min . Clearly if for some point q ∈ B , t ( q ) < t min , then q / ∈ ˜ J + [ E ]. Therefore, eachconnected component of B contains some points not in ˜ J + [ E ]. Therefore, if ˜ J + [ E ] ∩ B (cid:54) = J + [ E ∩ B ], then ∂ ˜ J + ( m ) ∩ B includes a hypersurface Σ in B that is not in ˜ J + [ E ∩ B ].We will now show that S cannot exist. – 28 –y lemma 10, E intersects B orthogonally. Therefore, in lemma 9, the secondtype of null geodesic in ∂ ˜ J + [ E ] does not exist. The first type of geodesic forms acodimension-two congruence starting orthogonally from E on which, except possibly atthe endpoints, every deviation vector is spacelike and orthogonal to the tangent vector.By lemma 5, each such geodesic either lies entirely in B or lies entirely in M except atits endpoints. In particular, the points in S must lie on geodesics that are entirely in M except where they end. We thus consider the congruence of geodesics in M startingorthogonally from E ∩ M . Since
E ∩ M is extremal, its expansion (with respect to g ab )is initially zero. By the null energy condition, its expansion is nowhere positive. Thusthe conditions of lemma 6 apply. Hence S consists of isolated points, contradicting thefact that it is a hypersurface in B . (cid:3) Note that theorem 7 is a special case of theorem 11, in which we take E to be asmall (in the metric ˜ g ab ) hemisphere centered on p and take the limit in which its radiusgoes to 0. Let Σ be a Cauchy slice of B . Given a codimension-zero submanifold of Σ, let A be itsinterior, ∂ A its boundary, and A c its complement; these three sets do not overlap andcover Σ. They naturally induce a causal decomposition of the spacetime B into fournonoverlapping regions (except that J ± [ ∂ A ] both include ∂ A ): Theorem 12 D [ A ] ∪ D [ A c ] ∪ J + [ ∂ A ] ∪ J − [ ∂ A ] = B (4.3) D [ A ] ∩ D [ A c ] = D [ A ] ∩ J ± [ ∂ A ] = D [ A c ] ∩ J ± [ ∂ A ] = ∅ (4.4) J + [ ∂ A ] ∩ J − [ ∂ A ] = ∂ A . (4.5) Proof: (4.4) and (4.5) are obvious from the definitions.We now prove (4.3). Suppose a point p ∈ J + [Σ] is not in any of the four regions.Each inextendible causal curve through p intersects Σ exactly once, but not in ∂ A (else p ∈ J + [ ∂ A ]). Nor can all such curves intersect it in A (else p ∈ D [ A ]) or A c (else p ∈ D [ A c ]). So some must intersect Σ in A and others in A c . Let λ be in the firstset and λ in the second. Join λ and λ at p to make a continuous curve λ from A to A c . Now, in any globally hyperbolic spacetime there exists a global timelike vectorfield; its integral curves can be used to construct a continuous map f from J + (Σ) toΣ. f ( λ ) is a continuous curve in Σ from A to A c . There therefore exists a point q ∈ λ – 29 –uch that f ( q ) ∈ ∂ A , and therefore q ∈ I + [ ∂ A ]. Since p ∈ J + ( q ), p ∈ J + [ ∂ A ], which isa contradiction. (cid:3) Now let E A be a surface in ˜ M that satisfies the conditions of theorem 11 and isspacelike-homologous to A . The precise meaning of the latter condition is as follows:There exists a Cauchy slice ˜Σ for ˜ M such that ˜Σ ∩ B = Σ, containing a codimension-zero submanifold with boundary A ∪ E A ; we call its interior R A . Since ˜Σ is itself amanifold-with-boundary (namely ˜Σ ∩ B ), one has to be careful about the definitionsof “interior” and “boundary” for a submanifold. We mean “interior” in the sense ofpoint-set topology; thus R A includes A but not E A . The “boundary” can be either inthe sense of “submanifold-with-boundary” (which is what we call ∂ R A ), or in the senseof point-set topology. In the latter sense, the boundary is just E A . As with A , wedefine R c A := ˜Σ \ ( R A ∩ E A ). To summarize, in parallel to the decomposition of Σ into A , A c , and ∂ A , we have a decomposition of ˜Σ into R A , R c A , and E A . Furthermore, R A ∩ B = A , R c A ∩ B = A c , and E A ∩ B = ∂ A .We can now apply theorem 12 to obtain a decomposition of ˜ M into the fourspacetime regions D [ R A ], D [ R c A ], J ± [ E A ]. The central result of this section is thatthis decomposition reduces on the boundary precisely to its decomposition into D [ A ], D [ A c ], and J ± [ ∂ A ]: Theorem 13 ˜ D [ R A ] ∩ B = D [ A ] (4.6a)˜ D [ R c A ] ∩ B = D [ A c ] (4.6b)˜ J ± [ E A ] ∩ B = J ± [ ∂ A ] (4.6c) Proof:
Equation (4.6c) is Theorem 11 (and its time reverse). Using Theorem 12 bothin B and in ˜ M to take the complement of both sides, we have (cid:16) ˜ D [ R A ] ∩ B (cid:17) ∪ (cid:16) ˜ D [ R c A ] ∩ B (cid:17) = D [ A ] ∪ D [ A c ] . (4.7)Lemma 1 then implies (4.6a), (4.6b). (cid:3) Theorem 13 immediately implies that E A is outside of causal contact with D [ A ]and D [ A c ], as required by field-theory causality.The spacelike-homology condition raises the following practical question: Given acodimension-one submanifold of ˜ M with boundary A ∪ E A , under what circumstances The point-set-topology boundary can be shown to equal the “edge” of the submanifold, in thesense used in the general-relativity literature (see e.g. [25]). – 30 –s it contained in a Cauchy slice? Obviously, it must be acausal. However, this isnot sufficient; for example, a spacelike hypersurface in Minkowski space that approacesnull infinity is not contained in a Cauchy slice. The following lemma, which will alsobe needed in theorem 15, shows that compactness is a sufficient additional condition.(This lemma applies in any globally hyperbolic spacetime.)
Lemma 14 If R is a compact acausal set, then there exists a Cauchy slice containingit. Proof: Let t ∈ R be a global time function, and define t max := max R ( t ), t min :=min R ( t ) (these exist since R is compact). Define Υ := { p : t > t max } and Υ (cid:48) :=Υ ∪ I + [ R ]. Define Σ := ∂ Υ (cid:48) = (cid:0) ∂ Υ \ I + [ R ] (cid:1) ∪ (cid:0) ∂I + [ R ] \ Υ (cid:1) . (4.8) ∂I + [ R ] contains R , and Υ ∩ R = ∅ , so R ⊂ ∂I + [ R ] \ Υ ⊂ Σ. Next we show that Σis achronal. The maximum value of t on Σ is t max , so there can be no future-directedtimelike curve from ∂ Υ to Σ. Further ∂I + [ R ] is itself achronal. Finally, if there is afuture-directed timelike curve from p ∈ ∂I + [ R ] to q ∈ ∂ Υ, then q ∈ I + [ R ] and hence q (cid:54)∈ Σ. So Σ is achronal.Next, we show that every inextendible future-directed timelike curve intersects Σ.On such a curve, t increases monotonically and continuously from −∞ to + ∞ . For t ≤ t min , the curve is not in Υ (cid:48) ; for t > t max , it is. Therefore for some value of t itintersects Σ.While Σ is achronal, it is not quite a Cauchy slice (in the sense used in this paper)because it is not acausal. However, since R is acausal, Σ can be deformed outside of R to be acausal. (cid:3) Theorem 15
Let Σ (cid:48) be a Cauchy slice for B and A (cid:48) ⊂ Σ (cid:48) a region such that A (cid:48) ∪ ∂ A (cid:48) is compact and D [ A (cid:48) ] = D [ A ] . Then A (cid:48) is spacelike-homologous to E A .Proof: Since E A and A (cid:48) ∪ ∂ A (cid:48) are both compact, E A ∪ A (cid:48) is compact as well.(Recall that ∂ A (cid:48) = ∂ A ⊂ E A .) E A and A (cid:48) are acausal, since each sits on a Cauchy slice.Furthermore, by theorems 12 and 11, there are no causal curves connecting them; hence E A ∪ A (cid:48) is acausal. Therefore, by theorem 14, there is a Cauchy slice ˜Σ (cid:48) containingboth E A and A (cid:48) .Choosing a global timelike vector field on ˜ M , its integral curves define a diffeo-morphism f : ˜Σ → ˜Σ (cid:48) . Let R (cid:48)A := f ( R A ). Since E A is contained in both Σ andΣ (cid:48) , f ( E A ) = E A . Since every timelike curve in D [ A ] intersects Σ in A and Σ (cid:48) in A (cid:48) ,– 31 – ( A ) = A (cid:48) . So R (cid:48)A := f ( R A ) is a region in ˜Σ (cid:48) with ∂ R (cid:48)A = A (cid:48) ∪ E A . (Strictly speaking,we also need to define a new Cauchy slice for B , Σ (cid:48)(cid:48) := ˜Σ (cid:48) ∩ B , and to consider A (cid:48) to bea region in Σ (cid:48)(cid:48) , since the equality Σ (cid:48)(cid:48) = ˜Σ (cid:48) ∩ B is part of the definition of the spacelikehomology condition.) (cid:3) Theorem 15 shows that the HRT formula gives the same value for the entanglemententropy of A and A (cid:48) , as required by field-theory causality. The main result of this paper, Theorem 13, shows that the HRT prescription for com-puting holographic entanglement entropy [32] is consistent with the requirements of fieldtheory causality. As we have explained with various simple examples and gedanken ex-periments in § a priori obvious, since there are severalmarginal cases where arbitrarily small deformation of the bulk extremal surface wouldplace it in causal future of a boundary deformation which however cannot affect theentanglement entropy. With the primary result at hand, we now take stock of thevarious physical consequences it implies for holographic field theories. Causality constraints on holography:
Let us start by asking what we can learnabout holography from causality considerations. Recall that we proved our result forextremal surfaces in the context of two-derivative theories of gravity satisfying the nullenergy condition. This was crucial for us to be able to use the Raychaudhuri equation inorder to ascertain properties of null geodesic congruences. Thus the domain of validityof our statements was strong coupling in a planar (large- N ) field theory. This translatesto demanding a macroscopic spacetime with (cid:96) s (cid:28) (cid:96) AdS in a perturbative string ( g s (cid:28) (cid:96) s , our conclusions will hold, since the domi-nant effect will come from the leading two-derivative Einstein-Hilbert term in the bulk.When the higher-derivative operators are unsuppressed we have little to say for tworeasons: (a) the holographic entanglement prescription so far is only given for staticsituations (or with time reversal symmetry) [43, 44] and (b) even assuming the co-variant generalizations, one is stymied by the absence of clean statements regardingdynamics of null geodesic congruences (even for example in Lovelock theories). One The family of f ( R ) theories can be brought to heel, since here we can map the theory to Einstein-Hilbert via a suitable Weyl transformation. Causality constraints can be discerned here so long as theWeyl transformation (which is non-linear in the curvature) is well-behaved. – 32 –ould, however, use the causality constraint to rule out certain higher-derivative theo-ries from having unitary relativistic QFT duals (see e.g. [45]); this is similar in spiritto the recent discussions on causality constraints on the three-graviton vertex [46].Turning next to 1 /N , or bulk quantum corrections, while we have less control ingeneral, we can make some observations about the leading 1 /N correction which hasbeen proposed to be given by the entanglement of bulk perturbative quantum fieldsacross E A [47]. Since the bulk theory itself is causal, it follows that entanglement acrossthe extremal surface satisfies the desired causality conditions. Does causality prove the HRT conjecture?:
One intriguing possibility given,the importance of the causality, is whether we can use it to constrain the location ofthe extremal surface in the bulk, and thus prove the HRT conjecture. Unfortunately,causality alone is not strong enough to pin down the location of the extremal surface.What we can say is that the extremal surface E A has to lie inside the causal shadow Q ∂ A . In a generic asymptotic AdS spacetime, for a generic region A , the casual shadowis a codimension-zero volume of the bulk spacetime M . It is only in some very specialcases that we zero in on a single bulk codimension-two surface uniquely (e.g., sphericalregions in pure AdS or in the eternal Schwarzschild-AdS black hole). Causality constraints on other CFT observables:
Our discussion has exclu-sively focused on the causality properties of a particular non-local quantity in the fieldtheory, namely the entanglement entropy. However, causality places restrictions onother physical observables we can consider on the boundary as well. For instance, cor-relation functions of (time-ordered) local operators, Wilson loop expectation values,etc., should all obey appropriate constraints which we can infer from basic principles.Indeed, this can be shown to be the case, for example, for correlation functions, byconsidering the fact that the bulk computation involves solving a suitable boundaryinitial value problem for fields in the bulk, which can be checked to manifestly satisfycausality.However, this is less clear when we approximate, say, two point functions of heavylocal operators using the geodesic approximation [48]. Similar issues arise for thesemi-classical computation of Wilson loop expectation values [49, 50] using the stringworldsheet area. In these cases, one generically encounters some tension between theuse of extremal surfaces—geodesics, two-dimensional worldsheets, etc.—for the bulkcomputation, and field theory expectations regarding causality (cf., [51] for an earlier We thank Vladimir Rosenhaus for inspiring us to think through this possibility. The examples are all cases where, by a suitable choice of conformal frame, the extremal surfacecan be mapped onto the bifurcation surface of a static black hole. The black funnel and dropletsolutions (see [30] for a review) provide nontrivial examples, cf., [23]. – 33 –iscussion of this issue). Indeed, it appears that codimension-two extremal surfaces arespecial in this regard, for we can rely on the boundary of the entanglement wedge beinggenerated by a codimension-one null congruence, and thus apply the Raychaudhuriequation. Understanding the proper application of the WKB approximation for otherobservables is an interesting question; we hope to report upon in the near future [52].
Entanglement wedges:
One of the key constructs in our presentation, naturallyassociated with a given boundary region A , has been the entanglement wedge W E [ A ].This is the domain of dependence of the homology surface R A (recall that R A formsa part of a Cauchy surface which interpolates between A and E A ). Equivalently, itcomprises the set of spacelike-separated points from E A which is connected to A , oneof the four regions in the natural decomposition of the bulk spacetime.Given A , one might ask how unique this decomposition is. Since W E [ A ] is acausally-defined set, its specification only requires the specification of the (oriented)extremal surface E A (possibly consisting of multiple components when so required bythe homology constraint). The prescription for constructing the null boundary of W E [ A ]is unambiguous: simply to follow all null normals (emanating from E A in the requisitedirection, towards D [ A ]) until they encounter another generator (i.e. a crossover seam)or a caustic. However, there is a possibility that the extremal surface itself is notuniquely determined from A . This happens when multiple (sets of) extremal surfacessatisfy (2.4) but have the same area. Since entanglement entropy itself cares only aboutthe area, the HRT (as well as RT and maximin) prescription is to take any of these.However, which we take does matter for the entanglement wedge. We propose that, justas for the extremal surfaces, in such cases we may have multiple entanglement wedges W E [ A ] associated to the same boundary region A .The most “obvious” class of examples where this can happen is the case of A consisting of multiple regions or in higher dimensions where the entangling surface ∂ A consists of multiple disjoint components. As we vary the parameters describingthe configuration, the extremal surfaces involved typically exchange dominance, soat some point their areas must agree. Applying continuity from both sides, at thetransition point, both entanglement wedges should be naturally associated with A .However, in complicated states, there can actually be multiple extremal surfaces evenfor when A and ∂ A are both connected. In such cases, we could have candidateentanglement wedges which are proper subsets of (rather than merely overlapping with)other candidate entanglement wedges.It is also interesting to note that the decomposition of the bulk into four spacetimeregions causally defined from E A need not coincide with the bulk decomposition definedfrom E A c , despite there being a unique boundary decomposition defined from ∂ A . For– 34 –FT R CFT L AQW E [ A c ] W E [ A ] Fig. 10:
Sketch of Penrose diagram for a symmetric Vaidya-Schwarzschild-AdS geometryobtained by imploding null shells to the past and future from both boundaries nowdisplaying the entanglement wedges and the causal shadow region, with A being a fullCauchy surface for CFT R . pure states, where the homology constraint trivializes and we have E A = E A c , we canwrite the bulk decomposition equivalently with respect to both A and A c , M = W E [ A ] ∪ W E [ A c ] ∪ ˜ J + [ E A ] ∪ ˜ J − [ E A ] (5.1)which is directly analogous to the boundary decomposition (2.1). However, for mixedstates, where typically E A (cid:54) = E A c , the decomposition (5.1) is not true; instead thecorrect decomposition should replace W E [ A c ] with the bulk domain of dependence ofthe complement of R A within the bulk Cauchy slice ˜Σ, or more precisely ˜ D [ ˜Σ \R A \E A ]. Dual of ρ A ? Within the class of CFTs and states with a geometrical holographicdual, it has often been asked, for a given region A , what is the bulk “dual” of thereduced density matrix ρ A . One way to formulate what one means by this is as follows:suppose we fix ρ A and vary over all compatible density matrices for the full state ρ . What is the maximal bulk spacetime region which coincides for all such ρ ’s? By Note however that if we purify a mixed state by additional boundaries, such as in the deformedeternal black hole example illustrated in Fig. 10, then the decomposition (5.1) does hold. In recent years this question has been invigorated by e.g. [53, 54]. – 35 –coinciding bulk regions” one means having the same geometry, i.e. the same bulkmetric modulo diffeomorphisms. Another way to define the dual of ρ A is to ask whatis the maximal bulk region wherein we can uniquely determine the bulk metric (againmodulo diffeomorphisms). In fact there are several (generally distinct) bulk regionsthat might be naturally associated with the density matrix; in nested order: • The bulk region that ρ A is sensitive to ; in other words, regions wherein a defor-mation of the metric affects ρ A . • The bulk region that ρ A determines , i.e. where we can uniquely reconstruct allthe components of the metric (up to diffeomorphisms). • The bulk region that ρ A affects , i.e. where by changing ρ A one can change thebulk metric.Here we focus on the second case, following [53, 54]. Based on lightsheet arguments,the authors of [53] proposed the causal wedge as the correct dual. On the other hand,[54], as well as [12, 22], argued that the requisite region should contain more than thecausal wedge. In particular, [54] presented a number of criteria that such a regionshould satisfy, and explored several possibilities, most notably the region they denotedˆ w ( D A ) which corresponds to the bulk domain of dependence of the spacetime regionspanned by all codimension-two extremal surfaces anchored within D [ A ]. If every pointof R A lies on at least one of these, then this region coincides with our entanglementwedge W E [ A ]. On the other hand, as [54] pointed out, there may be “holes” in such aset, i.e., regions of R A which do not lie along any least-area extremal surface anchoredon a given region A (cid:48) ⊂ A . We propose that, since the most “natural” causal set associated with ρ A from thebulk point of view is the entanglement wedge, this is indeed the most appropriate regionto be identified with the “dual” of the reduced density matrix ρ A (even in the presenceof such entanglement “holes”). In this context, we should note that we can strip awaythe rest of the boundary spacetime, and consider the field theory just on D [ A ], whichis a globally hyperbolic spacetime in its own right, in the state ρ A . Whether thisstate in general admits a holographic description is not known, but, if it does, then In fact there is a further subdivision here based on whether any geometrical deformation of themetric should change ρ A or merely whether there should exist some deformation of the metric whichchanges ρ A . We thank Mark Van Raamsdonk for discussions on this issue. The example given in [54] involves a region through which traversing surfaces are not the smallest-area ones anchored on the given region, but a simpler physical example would be a point sufficientlyclose to an event horizon of an eternal spherical black hole, with A = Σ of one side as considered in § – 36 – natural candidate would seem to be the entanglement wedge: this is, in its ownright, a globally hyperbolic, asymptotically AdS spacetime, whose conformal boundary(according to theorem 13) is precisely D [ A ], and the area of whose edge E A gives theentropy of ρ A .Here the word “natural” should be qualified, especially in light of the argumentsin [22] that the causal wedge W C [ A ] is a natural bulk codimension-zero region asso-ciated with A . The latter can be obtained more minimally: it suffices to know thecausal structure of the bulk to define W C [ A ]. On the other hand, the density matrixclearly encodes much more than the bulk causal structure, since at least it knows theentanglement entropy (as well as entanglement entropies of all subregions, apart fromother observables). Since, in the bulk, the corresponding extremal surface is definedonly once we know the bulk geometry, the entanglement wedge W E [ A ] it defines is aless minimal construct that the causal wedge W C [ A ]. Nevertheless, once E A is identi-fied, the rest of the bulk construction of the entanglement wedge is purely causal, andtherefore defined fully robustly for any time-dependent asymptotically AdS spacetime.The statement that the entanglement wedge is the natural dual of the reduceddensity matrix (which implies that the boundary observer in D [ A ] can learn aboutthe bulk geometry in the entire W E [ A ]) has a profound consequence. We have shownthat the extremal surface E A has to lie in the causal shadow. This set can however bequite large, and so E A can lie very deep inside the bulk (as indicated by the shadedregion in Fig. 10). In fact, a simple example supports the idea that the entanglementwedge represents the state in such a case (see Fig. 11). We start with a deconfinedthermal state at t = 0 on a single S d − , represented holographically by the exteriorSchwarzschild-AdS solution. We add an outgoing null shell that reaches the boundaryat t < t >
0. At t = 0 we still have the thermalstate. The bulk solution is also unchanged between the past and future shells. However,these shells move the singularity and therefore have the effect of bringing the futureand past event horizons closer to the boundary, leaving the previous bifurcation surfacehidden behind both horizons. While this surface is no longer the bifurcation surface ofa global Killing vector, it remains the extremal surface whose area gives the entropyof the state of the field theory on the right boundary. Presumably the holographicdescription of the state extends all the way down to this extremal surface, as it does inthe absence of the shells, and thus consists of the entire entanglement wedge.Another (related) example where the separation between entanglement wedge andcausal wedge is particularly striking is the eternal (two-sided) black hole deformed bymany shocks considered in [35, 55]. The Einstein-Rosen bridge is highly elongatedand the extremal surface probably lies somewhere in the middle of it—so that theentanglement wedge for the entire right boundary is substantially larger than the causal– 37 – AR A R A E A E A Fig. 11:
Left: Exterior AdS-Schwarzschild solution, dual to a deconfined thermal state on S d − .The extremal surface for the entire boundary (red dot) coincides with the bifurcationsurface and the causal information surface. Right: Vaidya solution with an ingoingnull shell that reaches the boundary at t < and an outgoing one that leaves it at t > (brown); the geometry between the shells is unchanged, but the past and futureevent horizons (blue) have moved closer to the boundary, leaving the extremal surface(red dot) hidden behind them. The entanglement wedge in both cases is the entirespacetime (with a homology surface shown in green), while the causal wedge in theright figure is just the part outside of the event horizons. (The causal informationsurface is shown as the black dot.) wedge, which in this case is simply the right exterior (domain of outer communication)of the black hole. So not only does the entanglement wedge penetrate arbitrarily closeto the curvature singularity, it also contains a substantial part of the spacetime farbeyond the black hole horizon! Acknowledgments
We would like to thank Hong Liu, Juan Maldacena, Don Marolf, Steve Shenker, MarkVan Raamsdonk, and Aron Wall for discussions. MH, VH, and MR would like to thankUniversity of Amsterdam for hospitality during the initial stages of this project. VH andMR would also like to acknowledge the hospitality of Brandeis University, MIT, Cam-bridge University, and the University of British Columbia, Vancouver during various– 38 –tages of this project. We would like to thank the Aspen Center for Physics (supportedby the National Science Foundation under Grant 1066293) for their hospitality duringboth intermediate and concluding stages of this project.MH is supported in part by the National Science Foundation under CAREERGrant No. PHY10-53842. VH and MR are supported in part by the Ambrose Monellfoundation, by the FQXi grant “Measures of Holographic Information” (FQXi-RFP3-1334) and by the STFC Consolidated Grant ST/J000426/1. AL is supported in partby DOE grant DE-SC0009987. MR is supported by the European Research Councilunder the European Union’s Seventh Framework Programme (FP7/2007-2013), ERCConsolidator Grant Agreement ERC-2013-CoG-615443: SPiN (Symmetry Principles inNature).
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