aa r X i v : . [ h e p - t h ] J u l Extremal Surface Barriers
Netta Engelhardt and Aron C. Wall
Department of PhysicsUniversity of California, Santa BarbaraSanta Barbara, CA 93106, USA [email protected], [email protected]
Abstract
We present a generic condition for Lorentzian manifolds to have a barrier that limits the reachof boundary-anchored extremal surfaces of arbitrary dimension. We show that any surfacewith nonpositive extrinsic curvature is a barrier, in the sense that extremal surfaces cannot becontinuously deformed past it. Furthermore, the outermost barrier surface has nonnegativeextrinsic curvature. Under certain conditions, we show that the existence of trapped surfacesimplies a barrier, and conversely. In the context of AdS/CFT, these barriers imply that itis impossible to reconstruct the entire bulk using extremal surfaces. We comment on theimplications for the firewall controversy. ontents K < Surfaces are Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 So are K = 0 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Trapped Surface Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Outermost Barriers have a K ≥ Direction . . . . . . . . . . . . . . . . . . . . . . 10
Despite the successes of the AdS/CFT correspondence, a full understanding of the duality isstill lacking. Since the original conjecture [1], much of the research effort has been aimed atdeveloping a precise dual dictionary connecting geometric quantities in the bulk to observables inthe boundary field theory. However, this dictionary remains poorly understood, hampering ourability to reconstruct the bulk geometry from the field theory, compute field theory quantitiesfrom the classical bulk theory, and assess whether or not the correspondence is complete. Thislast effort has been placed under scrutiny by the recent controversy over the black hole interior(see e.g.[2, 3, 4, 5, 6, 7, 8, 9, 10]), and direct efforts to construct – or demonstrate an inability toconstruct – a full dual dictionary, may serve to address it.To resolve such questions, a good starting point is the identification of bulk probes that dependprimarily on the bulk geometry and are dual by the AdS/CFT dictionary to known field theoryobservables. The existence of such bulk probes is not guaranteed; however, spacelike extremalsurfaces, which are covariantly-defined and depend exclusively on the bulk geometry, are dual towell-understood field theory observables. In fact, spacelike extremal surfaces constitute most ofthe probes used in AdS/CFT. The dual observables include correlators [11], entanglement entropy[12, 13], and Wilson loops [14]. The length of spacelike geodesics with boundary endpoints allowsone to compute, in the WKB approximation, the two-point correlator of a high conformal dimensionfield operator at the endpoints. The area of a codimension 2 spacelike extremal surface anchoredon some region R on the boundary of an asymptotically locally Anti-de Sitter (AlAdS) spacetimeis associated with the entanglement entropy of R within the boundary field theory.A field theory observable which is dual to an extremal surface must in some way encodeinformation about the bulk geometry at the location of the extremal surface. If there is a limit2n how far into the bulk such surfaces can reach, this also limits our ability to reconstruct thegeometry from the corresponding dual observables. The natural question that arises is how muchof the bulk can be recovered from extremal surface probes. Extremal surface probes and theirmaximal reach were studied in [15] for the case of static and translationally or spherically symmetricasymptotically AdS spacetimes , and in a large amount of literature (e.g. [17] and [18]) for eternalAdS black holes. Studies have also been done for extremal surfaces in time-dependent geometries(e.g. [13, 19, 20, 21, 22, 23]).In this paper, we establish a general constraint on the reach of probe spacelike extremal surfacesin spacetimes which are not necessarily AlAdS. We find that many generic geometries admit acertain kind of surface which acts as a barrier for probe surfaces anchored to one boundary.We furthermore show that trapped surfaces with nonpositive null extrinsic curvature give rise toextremal surface barriers, and that—on spacetimes which admit a totally geodesic spatial slice—theexistence of barriers implies the existence of either a singularity or a marginally trapped surface.More explicitly, we consider a codimension 1 “splitting surface” Σ which divides the spacetimeinto two regions. Σ may be spacelike, timelike, or null. We take some class X of spacelike extremalsurfaces of codimension 1 or greater, which are anchored to some boundary I on one side of Σ .Then, given two simple assumptions, we can show that no members of this class of extremalsurfaces cross (or even touch) Σ . In other words, the splitting surface acts as an extremal surfacebarrier. The two assumptions are as follows:1. All of the extremal surfaces in the class X can be continuously deformed (while remainingextremal) to surfaces which lie on the exterior of Σ .2. Σ has nonpositive extrinsic curvature (as measured by normal vectors pointing outward,towards the exterior of Σ ). In other words, Σ can only bend outward relative to its tangentplane.Given these assumptions, Σ acts as a barrier to any of the extremal surfaces lying in the class X . This generalizes some previously known result in Riemannian manifolds, that surfaces with aninward-pointing mean curvature vector can sometimes act as local barriers to minimal surfaces,and also act as global barriers in certain circumstances (see e.g. [24, 25, 26]).Our proof uses elements from the approach of [27] and [38], and shows that spacelike extremalsurfaces cannot approach surfaces with negative extrinsic curvature. They can, however, approacharbitrarily close to totally geodesic surfaces. We do not need to make any assumptions aboutcausality or an energy condition; the restriction is purely geometrical in nature.The first criterion is often satisfied: it simply requires that there be no obstruction to deformingthe extremal surface (while maintaining extremality) away from Σ . In many cases, this criterionholds for all extremal surfaces due to the topology and geometry of the spacetime, so the class A study on the reach of extremal surfaces in static black holes with higher derivative terms was done in [16]. For some field theory observables, the bulk extremal surface is required to be the minimal area surface of thosewhich are extremal. This can cause the location of the surface to jump discontinuously as one changes where it isanchored to I (e.g. geodesics in the BTZ black hole geometry [12]). This is not an issue for our proof, since we donot require that the deformed extremal surface be minimal. consists of all extremal surfaces. For example, this is true in vacuum AdS spacetimes. Moregenerally, if the extremal surfaces are anchored to just one connected component of the boundary I , and the bulk spacetime is homotopically trivial, then all surfaces may be continuously deformedtowards I . However, it may be an additional constraint on the geometry that the surfaces canremain extremal while this happens. For example, if there is another region Σ ′ which acts as abarrier to extremal surfaces anchored anywhere on the boundary of the spacetime ∂M , then Σ ′ may be an obstruction to deforming extremal surfaces past Σ .The second criterion may be satisfied by a totally geodesic surface, i.e. a surface with vanishingextrinsic curvature . An example of such a surface is a non–expanding black hole horizon (which,given the area theorem [28], is a stationary black horizon for spacetimes obeying the null energycondition). However, totally geodesic surfaces do not appear in generic spacetimes. Fortunately,the result also applies to surfaces with negative extrinsic curvature, which can appear in genericspacetimes. Examples will be given in Section 3. In fact, if there exists a surface Σ whose extrinsiccurvature is negative everywhere, we will show that it is not actually the tightest bound on extremalsurfaces. There will be some other “outermost” barrier Σ ′ some distance away, which has at leastpartly nonnegative extrinsic curvature (in Section 4, we will prove a theorem relating compactoutermost barriers to trapped surfaces and singularities).The existence of such extremal surface barriers and their link to singularities is a curiosity whichis of relevance to the firewall controversy. The fact that many bulk probes do not extend beyondthe barrier may suggest that the firewall, if it exists, may be at or behind the outermost barrier.At the very least, it shows that field theory observables that are dual to extremal surfaces anchoredat one boundary do not directly reveal any information about the interior. Any information aboutthis region in the bulk must come from probes that are not extremal surfaces. In some cases, thepresence of a barrier might even indicate that the boundary field theory does not have sufficientinformation to reconstruct the geometry behind the barrier [29]. In that case, additional factorsof the Hilbert space may be needed to describe that region.An interpretation involving a loss of determinism may at first seem implausible in the casewhere Σ is spacelike: one could use the bulk equations of motion to evolve the information in Σ forwards or backwards in time and thus reconstruct some or all of the interior of Σ . However,the presence of a firewall at Σ would result in a breakdown of the classical bulk equations ofmotion. In this case, our hope of reconstructing the interior of Σ relies entirely on the AdS/CFTcorrespondence dictionary. This dictionary, however, is primarily composed of extremal surfaceprobes, which we prove below cannot probe past Σ .This paper is structured as follows. In Section 2, we prove our sufficient condition for aspacetime to have a “barrier surface” past which extremal surfaces cannot reach; we further provethat trapped surfaces form extremal surface barriers. We will also show that the outermost barriersurface must have partly nonnegative extrinsic curvature. Section 3 provides examples of bulkspacetimes with extremal surface barriers. In Section 4, we prove that in a certain a class of A totally geodesic surface Σ can also be equivalently defined as a surface Σ in ( M, g ) such that every geodesicon Σ (with respect to the induced connection on Σ ) is also a geodesic on M (with respect to the connection on M . We first prove in two parts that extremal surfaces do not probe past surfaces with a nonpositiveextrinsic curvature tensor, and further, that barriers with everywhere negative extrinsic curvatureare not the outermost barrier surfaces. ( M, g ) will be a Lorentzian manifold with at least oneboundary I for the rest of this paper. We start with two definitions.Define a codimension 1 surface Σ in M to be a splitting surface if it separates M into two openregions, Ext (Σ) and Int (Σ) , where we define the interior and exterior relative to the normal whichdefines the extrinsic curvature, and ∂ Σ = ∅ . This normal is taken to point towards Ext (Σ) .Let Σ now be a splitting surface. Let { N r } be a family of spacelike extremal surfaces in M ofcodimension n ≥ such that all the N r can all be continuously deformed from some initial surface N ⊂ Ext (Σ) anchored at Ext (Σ) ∩ I , and all the N r are anchored at Ext (Σ) ∩ I (see, e.g. Fig.1). Then any surface in the family N r shall be called Σ -deformable . K < Surfaces are Barriers
Theorem 2.1:
Let Σ be a codimension 1 splitting surface in M such that for any vector field v µ on Σ , Σ K µν v µ v ν < , where Σ K µν is the extrinsic curvature of Σ . Any Σ -deformable spacelikeextremal surface which is anchored within I ∩ Ext(Σ) remains in Ext (Σ) . Moreover, no suchspacelike extremal surface ever touches Σ (or even comes arbitrarily close to touching Σ ). Proof.
The proof closely follows that of Lemma B in [27]. Let N r be the family of surfaces alldeformable from N as defined above. If all surfaces N r are in Ext (Σ) , we are done, since thatimplies that surfaces which are anchored at I ∩ Ext (Σ) can never be deformed past Σ .Suppose there exists a surface N I in { N r } such that N I ∩ Int (Σ) = ∅ . Then, since N I is linkedto N via a series of smooth deformations, there exists a “midway” surface N in the family ofsurfaces which coincides with Σ at a set of points { p i } and is tangent to Σ at those points. Fig. 1illustrates the deformation family. We focus on one coincident point p of Σ and N .As explained in [27], Σ K µν v µ v ν is a measure of how much Σ curves away from the tangent planenormal to Σ k µ with motion away from p in the v µ direction, where Σ k µ is the normal to Σ when Σ is not null, and we take Σ k µ to be the null generator for null Σ . Similarly, N K ρµν v µ v ν Σ k ρ measureshow much N curves away from the plane normal to Σ at p as one moves in the v µ direction away For simplicity, we will assume that all extremal surfaces in the family are twice differentiable, at least wherethey intersect Σ , so that the extrinsic curvature K µν is well-defined. Presumably this assumption can be weakenedif care is taken in dealing with distributional extrinsic curvatures. In terms of γ µν , the induced metric on Σ , the extrinsic curvature on Σ is given by Σ K µν = γ σµ γ λν ∇ σ Σ k λ . For asurface N of codimension n > , the extrinsic curvature carries an additional index, and is defined in terms of h µν ,the induced metric on N : N K ρµν = h σµ h νλ ∇ σ h ρλ . If Σ K µν v µ v ν > , we could prove, using precisely the same formalism used in the proof above, that surfacesanchored in Int (Σ) remain in Int (Σ) . I N N N I b p Figure 1: The family of deformations { N r } , all anchored at Ext (Σ) ∩ I . For Σ K µν v µ v ν ≤ , N I and N do not exist.from p (restricted to motion tangent to N ). N is outside of Σ , so Σ curves away from its tangentplane at least as much as N does: Σ K µν v µ v ν ≥ N K ρµν v µ v ν Σ k ρ (2.1)In particular: Σ K µν h µν ≥ N K ρµν h µν Σ k ρ = 0 (2.2) Σ K µν h µν ≥ (2.3)where h µν is the induced metric on N . The equality in Eq. 2.2 follows from the fact that N isextremal. N is spacelike, so h µν is Riemannian and has no temporal components, which implies that Σ K µν h µν < , since Σ K µν v µ v ν < . We therefore have a contradiction with Eq. 2.3, so the surfacesin the family { N r } never reach Σ , which in turn implies that N I does not exist. Moreover, it isimpossible for N to approach arbitrarily close to Σ , because in the limit Eq. 2.3 would still hold,and we would again arrive at a contradiction. K = 0 Surfaces
Theorem 2.2:
Let Σ be a splitting surface as above, but we now take it to be totally geodesic,i.e. Σ K µν = 0 . Any Σ -deformable spacelike extremal surface which is anchored in the intersectionof the boundary I with one component of M (as divided by Σ ) remains in that component . If we are only interested in barriers for extremal surfaces with a particular dimensionality, Eq. 2.2 may allowus to prove the existence of a barrier with a weaker condition than in Theorem 2.1. For example, to prove thata spacelike codimension 1 surface is a barrier for n -dimensional extremal surfaces, it is sufficient if the sum of thelargest n eigenvalues of Σ K µν are negative. However, the analysis is more complicated for timelike or null barriers. Suppose that M has some nontrivial topology or geometrical defect that creates an obstruction for Σ -deformability. If we can find another spacetime M ′ which agrees with M everywhere inside Σ and also obeysthe following conditions: (a) the extremal surface N is the same in M and M ′ , (b) the splitting surface Σ is thesame in M and M ′ , and (c) N is Σ -deformable in M ′ , then Σ is still a barrier in M , by the proof of Theorem 2.2. N b p Σ k µ ( p ) b q b s Σ k µ ( s ) Figure 2: A zoom-in near a neighborhood where Σ and N coincide. The horizontal lines at thepoint p represent the covector Σ k µ ( p ) = Σ k ν ( p ) g µν , where Σ k ν ( p ) is the null generator for null Σ ,or the normal for timelike or spacelike Σ . The horizontal lines at s represent the covector Σ k µ ( s ) ,which is obtained by parallel transporting Σ k µ ( p ) along Σ .Note that in the case where Σ is totally geodesic, it provides a barrier both for surfaces inInt (Σ) and Ext (Σ) . Proof.
The proof is identical for N ⊂ Int (Σ) and N ⊂ Ext (Σ) . For concreteness we assume N ⊂ Ext (Σ) .We take N to be the midway surface as above, and we show that if N agrees with Σ at p andis tangent to it, then N ⊂ Σ . This would imply that N is no longer anchored at I ∩ Ext (Σ) , andtherefore it is impossible to deform a surface to reach and cross Σ while maintaining boundaryconditions in Ext (Σ) . The saturated equation is: Σ K = N K ρ Σ k ρ = 0 (2.4)This shows that the two surfaces curve away in the normal direction equally, and is an indicationthat the two surfaces must agree everywhere on N if they coincide at a point. We now show thisin more detail.Suppose Σ is null, and let Σ k µ be the null generator of Σ directed towards N . The correspondingcovector is defined: Σ k µ = g µν Σ k ν , as illustrated in Fig. 2. Consider a small neighborhood U of p of length scale ǫ . Let q ∈ N ∩ U and let s ∈ Σ ∩ U . Define the coordinate y ( q ) : y ( q ) = Σ k µ ( s ) ( s µ − q µ ) (2.5)where Σ k µ ( s ) is obtained by parallel transporting Σ k µ ( p ) along Σ , and ( s µ − q µ ) is a vector pointingfrom s to p in the small neighborhood around p . Note that because Σ is totally geodesic, Σ k µ ( s )
7s null. Within U , we can always find an s ∈ Σ such that y ( q ) ≪ ǫ . For such a choice of s , wetherefore obtain: Σ k µ ( s ) − N k µ ( q ) = ∇ µ y ( q ) + O (cid:0) ( ∇ y ) (cid:1) (2.6)where N k µ ( q ) is obtained by parallel transporting N k µ ( p ) = Σ k µ ( p ) along N .The null extrinsic curvatures therefore obey: Σ K µν − N K ρµν Σ k ρ = ∇ µ ∇ ν y (2.7) N K ρµν Σ k ρ = −∇ µ ∇ ν y (2.8) N K ρ Σ k ρ = −∇ y (2.9) ∇ y (2.10)Let r , defined on Σ , be the proper distance from p . We can use r as a coordinate labeling pointson Σ . Let dσ be the volume element of a constant r slice. Define G ( x ) to be a Green’s functionon the ball of points x with r < R , where − ∇ G ( x ) = δ D − ( x ) ; G | r = R = 0 (2.11)where ∇ is defined on Σ . For sufficiently small R , the metric on Σ is approximately a flat Euclidean-signature metric, so G ∝ ( r D − − R D − ) / ( D − or ln( R/r ) in D = 4 . Since G ( x ) > (regardlessof what D is), we find ∂ r G | r = R < . When R is small, these inequalities must continue to hold ifthe metric is slightly deformed by nonzero curvature. Integrating over a ball B of radius r = R : Z B ∇ y G dr dσ = Z ∂ B y ∂ r G dσ (2.12)Since y is never negative (that would contradict the assumption that N does not cross Σ ), weconclude that y = 0 everywhere on a neighborhood of p . But this implies that y = 0 everywhere,since it solves an elliptic equation, and thus N ⊂ Σ . This completes the proof for null Σ . Thisreasoning applies directly to the cases where Σ is timelike or spacelike by simply substituting the N -pointing spacelike or timelike normal of Σ , respectively, for the null generator in the N direction.This proof allows us to generalize a previously-known result for totally geodesic surfaces: Corollary 2.3:
Let Σ be a totally geodesic codimension m surface in M . Let N be a codi-mension n ≤ m spacelike extremal surface satisfying the following conditions:1. N coincides with Σ at some point(s) p ,2. N does not cross Σ anywhere (and therefore N is tangent to Σ at p ),then N ⊂ Σ . 8 roof. Let Σ k µ i be the outwards-pointing normals to Σ as above (null generators in the N direction,if Σ is null), where i runs from , · · · , m . Let Σ k iµ = Σ k ν i g µν . Let U be a neighborhood of p . Wedefine y ( q ) for q ∈ N ∩ U as above: y i ( q ) = Σ k iµ ( s ) ( s µ − q µ ) (2.13)where s ∈ Σ ∩ U is chosen so that y i ( q ) ≪ ǫ . Then: Σ k iµ ( s ) − N k iµ ( q ) = ∇ µ y ( q ) i + O (cid:0) ( ∇ y ) (cid:1) (2.14)where for the case of null Σ , Σ k iµ ( p ) = N k iµ ( p ) , and for timelike and spacelike cases, we just requirethem to be equal. As before, we obtain Σ k iµ ( s ) by parallel transporting along Σ and N k iµ ( q ) byparallel transporting along N . The remaining reasoning in the proof of Theorem 2.2 above yields y i = 0 , and we are done.Note that Theorems 2.1 and 2.2, and Corollary 2.3 together also give a complete descriptionof the behavior of extremal surfaces in the presence of barriers, provided that all disconnectedcomponents are Σ -deformable. Corollary 2.3 also leads to a natural corollary of Theorems 2.1 and2.2, which combines the assumptions made in those theorems: Corollary 2.4:
Let Σ be a splitting surface in M such that Σ K µν v µi v νi = 0 for some vectorfields { v i } on Σ and Σ K µν u µi u νi < for all other vector fields { u i } on Σ . Then any Σ -deformablespacelike extremal surface which is anchored in Ext (Σ) remains in Ext (Σ) . Moreover, althoughsuch extremal surfaces might conceivably come arbitrarily close to Σ if they only propagate alongthe v i directions, they cannot approach Σ if they propagate along the u i directions. Proof.
As above, let { N r } be a family of Σ -deformable surfaces, all deformable from N ⊂ Ext (Σ) ,and let N be the midway surface between N I and N . We again focus on a coincident point p ∈ Σ ∩ N . By Eq. 2.3, we have Σ K µν h µν ≥ , where h µν is the induced metric on N . We candecompose the metric into the components that lie along the v directions and the u directions: h µν = h (1) µν + h (2) µν where h (1) µν = P i a i v i µ v i ν and h (2) µν = P i b i u i µ u i ν for some constants a i and b i . Eq. 2.3 can bedecomposed as follows: Σ K µν h µν = Σ K µν h (1) µν + Σ K µν h (2) µν (2.15)If N has no components in the u i directions, then h (2) µν = 0 , and Eq. 2.15 reduces to Σ K µν h µν = Σ K µν h (2) µν = 0 . This is simply the case of Theorem 2.2, so N could potentially get arbitrarily closeto Σ while remaining on Ext (Σ) .If N has at least one component in u i directions, then Eq. 2.15 yields Σ K µν h µν = Σ K µν h (1) µν + Σ K µν h (2) µν = Σ K µν h (2) µν < . This is simply the case of Theorem 2.1, so N cannot approach Σ ,and must always remain in Ext (Σ) .We conclude that if N propagates in the directions in which Σ has vanishing extrinsic curva-ture, N can approach Σ , but if N propagates in the directions in which Σ has negative extrinsiccurvature, N cannot approach Σ . 9 .3 Trapped Surface Barriers One consequence of Corollary 2.4 is that any null splitting surface Σ which is foliated by surfaceswhere all components of the null extrinsic curvature are nonpositive is an extremal surface barrier.But if we are only interested in whether or not Σ is a barrier to codimension 2 extremal surfaces,it turns out the we can do better: it is sufficient if the expansion θ ≤ , i.e. the trace of thenull extrinsic curvature is nonpositive. This is because when a codimension 2 extremal surface N touches (but does not cross) Σ , we can take the trace of Eq. 2.1 over all D − spacelike directionsto obtain: ≥ θ Σ ≥ θ N = 0 . (2.16)As in the case of Corollary 2.3, the above equation can only be saturated if N lies on Σ . Thismeans that Σ is a barrier to Σ -deformable spacelike extremal codimension 2 surfaces.If we assume the null curvature condition R µν k µ k ν ≤ , we can additionally show how to usea codimension 2 extremal surface to construct a barrier to other codimension 2 surfaces. Theorem 2.5:
Let X be a codimension 2 spacelike extremal surface. Let Σ be the union ofnull congruences shot outwards from X towards both the future and the past directions. Assumingthe null curvature condition, Σ is a barrier to Σ -deformable codimension 2 surfaces anchored toExt( Σ ). Proof.
By the standard construction, the Raychaudhuri equation implies that the two null con-gruences shot out from X converge, so that θ Σ ≤ , where θ is defined as the expansion movingoutwards away from X . As in the proof of Theorems 2.1 and 2.2, we assume for contradictionthat N is a midway surface touching Σ but not crossing it. (It does not matter which of the twonull congruences N touches first, but whichever one it touches first, we are only interested in the θ of that congruence.) At the point of coincidence, Eq. 2.3 implies that θ Σ and θ N both vanish.Therefore, by the same reasoning as in Corollary 2.3, N must lie entirely on Σ . But then it cannotbe anchored at Ext( Σ ).Even without assuming the null curvature condition, the following Corollary follows immedi-ately: Corollary 2.6: If Σ is a null surface foliated by (marginally) outer trapped surfaces (i.e. θ ≤ ), then Σ is a barrier to Σ -deformable codimension 2 surfaces anchored to Ext( Σ ).Thus, if we are only interested in codimension 2 extremal surfaces (e.g. for purposes of holo-graphic entanglement entropy [12, 13]), we can prove the existence of barriers in more circum-stances. K ≥ Direction
We proved that the presence of the surface Σ with nonpositive spatial extrinsic curvature is asufficient condition for an extremal surface barrier. This raises the question of whether this barrieris the closest one to the extremal surfaces. Consider some region on the boundary of the spacetime,10 ⊂ I . Let X i denote the spacelike extremal surfaces anchored at R . We define the outermostbarrier Σ for surfaces anchored at R to be given by : Σ = ∂ [ i X i ! . (For some (non-AlAdS) spacetimes, R may be a proper subset of Ext (Σ) , as shown in Fig. 3.)The above definition is partly motivated by [30], although the case under discussion there involvedonly minimal area surfaces, whereas we consider all extremal surfaces. Note that we can similarlydefine the outermost barrier for a subset { Y i } of all spacelike extremal surfaces anchored at R bydefining Σ Y = ∂ (cid:18)S i Y i (cid:19) . R Σ Figure 3: A illustration of a possible situation in which the extremal surface (in red) are anchoredat R , but they give rise to an outermost barrier Σ (in blue above) whose exterior contains R as aproper subset.We now prove some properties of the outermost barrier. Theorem 2.7:
Let Σ be an outermost barrier in M , and let { N r } be a family of extremalsurfaces that can all be smoothly deformed from some initial surface N ⊂ Ext (Σ) such that somesurface N ∈ { N r } is arbitrarily close to Σ . Then Σ K µν h µν ≥ , where h µν is the induced metric on N . Proof.
By definition of outermost barrier, for any point p , spacelike extremal surfaces in Ext (Σ) can be deformed to either come arbitrarily close to Σ or coincide with and be tangent to Σ at p . In the latter case, we can simply take the limiting surface that touches Σ . In any smallneighborhood where Σ and one of these spacelike extremal surfaces touch or nearly touch, Eq. 2.3yields Σ K µν h µν ≥ , where h µν is the induced metric on the extremal surface, and we are done.In particular, the extremal surface is spacelike and h µν is Riemannian, so in order for all of thecomponents of Σ K µν to add up to a nonnegative number, at least one eigenvalue of Σ K νµ must benonnegative.Whenever we can find a barrier of entirely negative extrinsic curvature, we can therefore find abarrier of partly nonnegative curvature. This is in agreement with Theorem 2.1, which implies that11xtremal surfaces cannot come arbitrarily close to a surface with strictly negative curvature. Theexistence of a barrier is therefore quite generic, since we expect it to occur for any surface Σ whichsplits the spacetime into 2 regions and has negative or vanishing extrinsic curvature components.Many of the known analytic solutions that are AlAdS admit a totally geodesic surface thatacts an as extremal surface barrier, as well as multiple surfaces with negative extrinsic curvature.Totally geodesic surfaces are quite special and therefore spacetimes containing them are not repre-sentative of general AlAdS spacetimes. We expect, however, that small perturbations of spacetimesthat admit splitting surfaces with negative extrinsic curvature do again result in spacetimes withsplitting surfaces of nonpositive extrinsic curvature. The existence of such surfaces is thus stableunder small perturbations of the metric.We therefore conclude that many generic AlAdS spacetimes admit an extremal surface barrier .Moreover, we show in the examples below that the barrier often separates a region of proximity toa singularity from the boundary of the spacetime. In Section 4, we further prove that for certainspacetimes, the existence of a compact outermost barrier implies the existence of singularities.The implications of this barrier in the context of AdS/CFT are disturbing: as rule of thumb, thebest-understood aspect of the duality dictionary is generically limited in the scope of informationit contains about the bulk. This at best suggests that we must change our approach towardsextracting bulk information by using probes that are not extremal surfaces. At worst indicatesthat complete information of a large class of bulk geometries may simply not be contained in thedual field theory [29]. In this section, we provide some examples of spacetimes with barriers. Some of these barriers aretotally geodesic, others have negative extrinsic curvature. The latter condition is stable undersmall perturbations, although the former is not.
The simplest example is pure AdS itself. If we consider the metric on the Poincaré patch: ds = 1 z (cid:0) − dt + dx i + dz (cid:1) (3.1)Surfaces of constant t or surfaces of constant x i are totally geodesic. Since extremal surfaces in pureAdS remain on constant time slices, it is clear that they do not cross constant t totally geodesicsurfaces. Similarly, a surface which is anchored between x i = − x and x i = x will not propagateto x i > x or x i < − x . The next example we consider is an isotropic AdS cosmology. The metric on global AdS can bewritten, via a coordinate transformation (see e.g. [19]), as an open Friedmann-Robertson-Walker12FRW) universe in the interior of the lightcone, and foliated into de Sitter slices in the exterior ofthe lightcone: ds int = − dt + sin ( t ) (cid:0) dχ + sinh χ d Ω (cid:1) (3.2) ds ext = dr + sinh ( r ) (cid:0) − dτ + cosh τ d Ω (cid:1) (3.3)where the two are related via an analytic continuation (see Fig. 4). Any small perturbation tothis metric results in a curvature singularity replacing the coordinate singularity. If we couple themetric to, say, a scalar field, as in [31], the resulting geometry features a curvature singularity (thefirst indication of some sort of an accumulation surface for codimension 2 surfaces in this geometrywas found in [19]): ds int = − dt + a ( t ) (cid:0) dχ + sinh χ d Ω (cid:1) (3.4) ds ext = dr + b ( r ) (cid:0) − dτ + cosh τ d Ω (cid:1) (3.5)where a (0) = 0 and b (0) = 1 at the lightcone and a ( t singularity ) = 0 is a big-crunch collapse. It wasfound in [32] that this spacetime contains a totally geodesic barrier: there is a point t m at which a ( t m ) reaches a maximum. The extrinsic curvature of any constant t surface with respect to thepast-pointing timelike normal is given by: K ij = a ′ ( t ) g ij where g ij is the FRW metric inside the lightcone. The constant t = t m surface Σ is therefore totallygeodesic, and surfaces with t > t m have negative extrinsic curvature. We demonstrate the barrierat work here using spacelike radial geodesics as an example of extremal surface probes. This iseasily generalized to extremal surfaces with fewer symmetries, but for brevity we limit ourselvesto the simplest case.Consider a geodesic with endpoints at θ = 0 and θ = π , where θ is one of the suppressed anglesin Eq. 3.5. The length functional for this geodesic within the FRW region is: L = Z p − dt + a ( t ) dχ = Z p − t ′ ( χ ) + a ( t ( χ )) dχ (3.6)where we have chosen χ to parametrize the geodesic. The boundary conditions θ = 0 and θ = π imply t ′ ( χ = 0) = 0 . We solve for the geodesic from the lightcone interior outwards, startingfrom χ = 0 to the lightcone and the exterior region, using the reflection symmetry in θ along thegeodesic path.Since χ is a cyclic coordinate, the Hamiltonian is conserved: H ( t ) = − a ( t )( a ( t ) − t ′ ( χ ) ) / = H ( t ( χ = 0)) = a ( t ( χ = 0)) . (3.7)Solving for t ′ ( χ ) yields: t ′ ( χ ) = ± a ( t ( χ )) − (cid:18) a ( t ( χ )) a ( t (0)) (cid:19) ! / . (3.8)13 RW d +1 Σ dS d Figure 4: The isotropic AdS d +1 cosmology. Σ is the surface where a ( t ) is maximized.Since the scale factor vanishes at the lightcone and at the singularity, and reaches its maximum atsome intermediate time t = t m , it follows from Eq. 3.8 that any geodesic with t ( χ = 0) > t m mustpropagate towards progressively larger t . Geodesics with t ( χ = 0) = t m stay on t ( χ ) = t m for all χ . This is precisely what Theorem 2.2 states: extremal surfaces in the interior of Σ (i.e. t > t m )remain in the interior, extremal surfaces in the exterior of Σ (i.e. t < t m ) remain in the exterior,and extremal surfaces that coincide with Σ (i.e. t = t m ) lie on Σ , where Σ is the totally geodesicsurface given by t = t m .One implication of this result is that, in the isotropic AdS Cosmology, only geodesics on theexterior of Σ make it to the lightcone at t = 0 and from there to the boundary. The totally geodesicsurface prevents boundary-anchored probes from getting arbitrarily close to the singularity. Another prominent example of a spacetime with a totally geodesic barrier surface as prescribedin Section 2 includes any black hole with a stationary horizon [33, 34] . This category includesAlAdS spacetimes such as Schwarzschild-AdS, RN-AdS, Kerr-AdS, the BTZ black hole, the planarAdS black hole (the horizon in this case was separately proven to be a barrier surface in [15]),and the hyperbolic AdS black hole. In particular, in AdS-Schwarzschild, there are two totallygeodesic surfaces: the past- and future-directed horizons, denoted Σ and Σ in Fig. 5. Spacelikeextremal surfaces anchored at I must therefore always remain in region 1, while those anchoredat I must remain in region 3 of the spacetime. Extremal surfaces anchored at both I and I are not constrained by our theorems to remain in any part of the spacetime. In fact, [17] foundthat spacelike geodesics with one endpoint at I and one endpoint at I can probe arbitrarilyclose to the singularity, and in particular, can cross both Σ and Σ . We emphasize that this Stationary geons, which are not black holes, also exhibit a barrier at the horizon. I I Σ Σ Figure 5: The Schwarzschild-AdS d +1 black hole. Σ and Σ are totally geodesic splitting surfaces,and are therefore by Theorem 2.2 extremal surface barriers.does not contradict our results in the previous section because these geodesics are anchored totwo boundaries. As a point of interest, [18] found a barrier surface, or accumulation surface,for codimension 2 extremal surfaces anchored to both boundaries, while, as noted above, radialgeodesics observe no such barrier .It is clear at this point that spacetimes with singularities seem to have a particular proclivityfor admitting barriers. This is not a coincidence: Corollary 2.6 states that null surfaces foliated by(marginally) trapped surfaces act as barriers at least for codimension 2 extremal surfaces. If weadditionally assume the stronger statement that all components of the null extrinsic curvatureare nonpositive, we found in Corollary 2.4. that it is a barrier to extremal surfaces of everydimension.In particular, when the null barrier Σ is spatially compact, i.e. the entire boundary is in Ext (Σ) ,then Σ is ruled by trapped surfaces. If the spacetime is furthermore globally (or AdS) hyperbolic,spatially noncompact, and obeys the null curvature condition, the existence of trapped surfacesguarantees the existence of singularities and horizons [36]. In the next section, we will prove apartial converse of our conclusions from Corollary 2.4: barriers imply the existence of trappedsurfaces or singularities, at least for spacetimes which admit a totally geodesic spacelike slice.It is worthwhile here to comment on any bearing this might have on the recent controversy overthe completeness of the AdS/CFT correspondence. The question of whether the black interior isfully described by the boundary field theory is particularly relevant to this discussion. In [3, 4, 8], The Vaidya spacetime, which we discuss in more detail in Sec. 5, has just one boundary, does not have a barrierfor all extremal surfaces because spacelike geodesics which enter at early times can leave at arbitrarily late times.However, there is an effective barrier for surfaces that are anchored to the boundary at late times, and there maybe a barrier for codimension 2 surfaces. Following the first version of this paper, an analysis of the behavior ofextremal surfaces in some Vaidya spacetimes was done in [35]. If we assume spherical symmetry, this condition automatically follows for trapped surfaces.
15t was shown that there exist operators in the static black hole interior that cannot exist in thedual field theory. The fact that a well-used probe of the bulk geometry cannot reach into a bulkregion which is not completely described by the boundary field theory may not be coincidental.Since the limited reach of extremal surfaces in the static black hole geometry is simply a specialcase of the barrier theorems presented in Section 2, we may expect that other black hole AlAdSgeometries admitting barriers may manifest similar incompleteness on the dual field theory side.One may also speculate that field theories dual to other geometries, not necessarily black holes,may exhibit the same behavior near a singularity.
The examples in the previous section suggest that there is some connection between singularities(particularly singularities masked by horizons), and extremal surface barriers. In this section, wewill prove this result in some special cases. Consider a spacetime manifold M containing a totallygeodesic spacelike slice S . Such slices can be found, for instance, in static spacetimes, or moregenerally, spacetimes with a moment of time reflection symmetry. Since we intend to drop inextremal surfaces from the boundary of S , we must assume that S is noncompact. Because S istotally geodesic, these extremal surfaces remain on S .Suppose further that there is an outermost barrier Σ such that the intersection between Σ and S is nonempty and compact. Then we prove below that S either admits a singularity, or else amarginally trapped surface.If, in addition, we assume global hyperbolicity (including AlAdS spacetimes with the appro-priate generalization of global hyperbolicity), the null curvature condition ( R µν k µ k ν ≥ for allnull vectors k µ ) and the generic condition (roughly, that each null ray encounters at least a littlebit of null curvature or shear) then the existence of a marginally trapped surface itself implies theexistence of a singularity somewhere on M [36].We first prove a lemma: Theorem 4.1:
No compact extremal surface barrier Σ in M ever touches extremal surfacesanchored to I . Proof.
We prove this by contradiction. Let Σ be a barrier, and suppose there exists a spacelikeextremal surface N which is anchored to I . By assumption, N ⊂ Ext (Σ) everywhere except whereit coincides. In any neighborhood of such a coincident point p , there exists a point such that: Σ K µν h µν > N K ρµν h µν Σ k ρ = 0 (4.1)where Σ k ρ is the null generator of Σ for null Σ , and the N -directed normal to Σ otherwise. N cannot lie on Σ for a continuous neighborhood (otherwise it would lie on Σ everywhere,which would violate the boundary conditions on N ). N can therefore touch Σ while remaininganchored on the boundary. But then we can slightly deform N in a neighborhood of p , to make anew surface N ′ which crosses Σ . Using the elliptical equation of extremality, we can solve for N ′ s N S M Σ Figure 6: A spacetime M with a timelike barrier Σ , projected onto a spacelike codimension 1totally geodesic slice S yields a compact barrier Σ S for extremal surfaces on S .outside the neighborhood as well. Because the perturbation is small, N ′ must still be anchored tothe boundary. This shows that Σ is not a barrier, and we have arrived at a contradiction.Theorem 4.1 may at first seem to indicate that compact outermost barriers cannot exist, sincewe have defined outermost barriers as the boundary of the union of some set of extremal surfaces:one might naturally ask how the outermost barrier is constructed, if extremal surfaces cannot touchit. The barrier must be constructed as a limit of extremal surfaces that come arbitrarily close totouching it. We assume therefore, that one can find sequences of extremal surfaces approachingany point p , whose limit is an extremal surface tangent to Σ at p . However, the limiting extremalsurface must not be anchored anymore to the boundary, or it would contradict Theorem 4.1. Wewill make use of this limit construction of the outermost barrier in the proof of Theorem 4.2 below,where we prove a condition relating the existence of barriers to the presence of singularities andtrapped surfaces.Because we are taking a limit of extremal surfaces, one may worry that this limit will berelated to the boundary in a bizarre way, perhaps by spiraling around, getting arbitrarily close tothe boundary without actually being anchored to it. We will deal with these pathological cases byincluding them in the following definition: a surface N is weakly anchored if there exists a d ∈ R such that all points p ∈ N are less than distance d away from the boundary I along N (aftercompactification of I ). Theorem 4.2:
Let M admit a totally geodesic codimension 1 spacelike slice S . Given acompact, nonempty, outermost barrier Σ S for weakly-anchored, codimension 2, Σ S -deformable (i.e.where the deformations are restricted to S ) spacelike extremal surfaces on S (e.g. Fig. 6), thenone of the following is true: We assume that all points on S are a finite distance away from the boundary after compactification.
17. There exists a singularity outside or on Σ S , or2. Σ S is a marginally trapped surface (in either time direction). Proof.
Since Σ S is an outermost barrier, there exist extremal surfaces that either coincide withit and are tangent to it at coincident points, or come arbitrarily close to coinciding with it. Theformer case is ruled out by Theorem 4.1.Let { N r } be a family of spacelike extremal surfaces that approach arbitrarily close to somepoint q ∈ Σ S while remaining weakly anchored on I S = I ∩ S . In a neighborhood of q , we take thelimit of these surfaces N ′ of { N ′ r } that reaches Σ at q .Since we are operating at the “physics” standard of rigor, we assume without proof that (a) thislimit must exist for some sequence of extremal surfaces, and that (b) the resulting extremal surfacemay be extended outside the neighborhood of q by solving the elliptical equation of extremality.When we solve this equation, by Theorem 4.1, N ′ must no longer be weakly anchored on I S .Since N ′ cannot be weakly anchored to I S , part of it must terminate somewhere in the interiorof the spacetime. It can do it in one of the following two ways (Fig. 7): (a) N ′ terminates in a singularity on or outside Σ S , or (b) N ′ spirals outside Σ S forever, without ever coming to an end. In that case, we can define N ′∞ to be the limit set of the extremal surface as it gets arbitrarily far away from the pointswhere it is anchored to the boundary. N ′∞ must be bounded by some compact innermostsurface Σ ′ S , where Σ ′ S ∩ I = ∅ . If (a) is true, we get the first part of the claim. We now show that (b) results in the second partof the claim. To do so, we need to show that Σ S = Σ ′ S . We do this by contradiction.Suppose Σ ′ S = Σ S . Then Σ ′ S ⊂ Ext (Σ S ) . This means that it is not a barrier for extremalsurfaces weakly anchored at I , and therefore by Theorem 2.1 there must exist a point p ∈ Σ ′ S suchthat tr S (cid:16) Σ ′ S K µν (cid:17) | p > . (4.2)On the other hand, by construction, N ′∞ ⊂ Int (Σ ′ S ) can come arbitrarily close to Σ ′ S at anypoint p ∈ Σ ′ S . This implies that the following holds everywhere:tr S (cid:16) Σ ′ S K µν (cid:17) ≤ . (4.3)(The proof is essentially the same as Theorem 2.7, but reversing the roles of the interior andexterior.) If N ′ is not (weakly) anchored to the boundary anywhere, then N ′∞ = N ′ . Note that N ′∞ cannot come arbitrarily close to the boundary, since then it would be boundary-anchored by thegeneralized definition above. NN’ Σ s Σ s ’ (a) SNN’ Σ s Σ s ’ (b) Figure 7: (a) The original extremal surface N and its limiting surface N ′ , in the case where N ′ terminates at a singularity, (b) The extremal surfaces N and N ′ , where N ′ spirals into somelimiting surface bounded by Σ ′ S .We have arrived at a contradiction, so instead we must assume that Σ ′ S = Σ S . This tells usthat N ′ must in fact coincide with Σ S . Since N ′ is extremal, it follows that Σ S satisfiestr S (cid:0) Σ S K µν (cid:1) = 0 . (4.4)This is the same as saying that Σ S is a marginally trapped surface, which proves the second partof the claim.The assumption that there exists an outermost barrier Σ S for codimension 2 extremal surfacesliving on S is reasonable: if we take M to have a totally geodesic slice S and a compact outermostbarrier Σ such that Σ ∩ S = ∅ , then in general, we expect that Σ ∩ S will be compact, and bea barrier for extremal surfaces on S . There should then exist some outermost Σ S which, at leastfor AlAdS spacetimes, will still be compact. (If there is a barrier for all extremal surfaces, thenobviously there must be a barrier for those which are codimension 2 and restricted to S .)If we further assume global hyperbolicity, the null curvature condition, and the generic con-dition, we are able to say that marginally trapped surfaces only occur in geodesically incompletespacetimes, so that barriers in these spacetimes exist only in the presence of singularities [36].Singularities are therefore intrinsically linked to extremal surface barriers. Assuming some formof cosmic censorship, these singularities must be hidden behind horizons, thus outermost barriersare also linked to horizons.In particular, the direct application of Theorem 4.2 to the AdS/CFT correspondence suggeststhat in general, extremal surface probes are not good probes for spacetimes with singularities.We expect that Theorem 4.2 can be further generalized to both the noncompact barrier case,as Section 3.2 suggests, and to any time-dependent geometry, although this may require assumingthe null curvature condition (implied by the null energy condition and Einstein’s equations), evento prove the existence of the trapped surface. For example, the proof of strong subadditivity for extremal surfaces can be proven without the null curvature Discussion
We proved in Theorem 2.1, Theorem 2.2, and Corollary 2.4 that a splitting surface with nonpositiveextrinsic curvature acts as a barrier to extremal surfaces (at least those which are Σ -deformable).This can include spacelike, timelike, or null barrier surfaces. If we are only interested in barriersto codimension 2 surfaces (as used in the holographic entanglement entropy conjecture [12, 13]),then we have shown in Theorem 2.5 and Corollary 2.6 that null surfaces foliated by (marginally)trapped surfaces are barriers.Conversely, for spacetimes with a totally geodesic slice, the existence of a barrier guaranteesthe existence of either singularities or else trapped surfaces for spacetimes with a totally geodesicslice S , so long as the intersection of the barrier and S is compact. We also showed that outermostbarriers have at least one nonnegative extrinsic curvature component (Theorem 2.7), and we arguedthat extremal surface barriers occur somewhat generically in AlAdS spacetimes. This presents asetback to attempts to reconstruct the bulk geometries from field theory operators using extremalsurface probes.These results are particularly troubling in light of how pervasive extremal surface probes are inapplications of the AdS/CFT correspondence. If there are regions which cannot be reconstructedfrom the entire boundary, than either those regions do not really exist as semiclassical regions (e.g.because there is a firewall at their boundary), or else if they do exist, the information inside ofthem must be contained in a new factor of the Hilbert space which is in addition to the boundary[29].We have also studied cases in which there is a barrier to extremal surfaces located on only part of the boundary. In this case it is not surprising that the entire bulk cannot be reconstructed.Thus the location of the barrier might give clues about how much of the bulk can be reconstructedfrom a CFT region. See [30, 38, 39, 40] for discussion of this question.One should bear in mind that extremal surfaces are not the only probe used in AdS/CFT. Ifthere are other bulk probes besides extremal surfaces, it is possible that a firewall (if it exists at all)might be located somewhere behind the outermost barrier. Conversely, if there are some extremalsurfaces which do not actually correspond to dual observables (for example, because they are notthe minimum area extremal surface), then it could conceivably be that the firewall is actuallyoutside of the barrier. It may also be that there are regions of space where some kinds of extremalsurfaces can probe, but not enough to fully reconstruct the geometry. It is unclear what the statusof these regions would be.In the case of a Vaidya spacetime where a black hole forms from the collapse of a shell, thereis no extremal surface barrier. In particular, there are spacelike geodesics which pass through anypoint of the black hole interior. The AMPS argument for firewalls [2] applies to sufficiently oldblack holes that form from collapse, yet the absence of a barrier might seem to suggest that therecan be no firewall. However, the spacelike geodesics in question are anchored at very large timeseparation on the boundary. In fact, [41] argued that the WKB stationary phase approximationis not dominated by spacelike geodesics passing through the interior of the black hole, but rather condition in the static case [37], but requires it in the dynamical case [38].
20y complex geodesics. It is unclear how to use the barrier results proven here in the case wherethe extremal surfaces are complex, as was found in [17] for two-point correlators in the case of anAdS-Schwarzschild bulk.However, even for the Vaidya spacetime, we still find that there is a partial barrier, for spacelikesurfaces anchored sufficiently far to the future of the collapsing shell. This is because the spacetimeis just Schwarzschild at late times. Thus the extremal surface barrier might still give clues aboutthe location of the firewall, although it does not necessarily tell us when the firewall would firstappear.In general, our knowledge of the AdS/CFT dictionary comes from a bootstrapping procedurewhere (a) proposed new ingredients to the dictionary can be checked using our knowledge of bulkphysics, while (b) the behavior of the bulk can also be predicted using the dictionary. But if atstep (b), our current understanding of the dictionary predicts a firewall—resulting in a breakdownof the bulk equations of motion used at step (a)—then it is unclear how to proceed. Should wemodify the bulk equations of motion or the dictionary? Perhaps it is the local bulk equationsof motion which should be kept sacrosanct, and other things we think we know about AdS/CFTshould be modified in order to preserve them. Ultimately, the question is what choice of dictionaryleads to the most consistent, complete, and interesting form of the correspondence.There are several remaining questions worth noting. First, we have yet to find a completedescription of the most general barrier, in particular for the outermost barrier surface. Such adescription would facilitate an understanding of precisely how generic extremal surface barriersare. We expect that, assuming the null curvature condition, it should be possible to show thatcompact outermost barriers can only occur in spacetimes with singularities.We note that (aside from Theorem 2.5) the main results of this paper are purely geometrical innature—they apply to any Lorentzian spacetime without needing to use of any energy conditionrestricting the sign of the curvature. (Although given the existence of trapped surfaces as shownin Theorem 4.2, the null curvature condition would say that there must be singularities.) Thissuggests that our results are still valid even in semiclassical spacetimes (e.g. evaporating blackholes) where local energy conditions are violated. However, the significance of these geometricalresults for AdS/CFT is not the same, since in the semiclassical (finite N ) regime, there are quantumcorrections to the dictionary relating extremal surfaces to boundary physics.For example, the entanglement entropy of a CFT region is no longer given just by the areaof an extremal surface. Instead it receives corrections due to the bulk entanglement entropy [42].Thus we should really be interested in surfaces which extremize the area plus the bulk entangle-ment entropy. To find barriers to these surfaces, it is natural to look for null surfaces foliated byquantum trapped surfaces, where the sum of the area and the bulk entropy is decreasing. One canthen prove quantum generalizations of the types of classical theorems proven here [27].21 cknowledgements We thank Dalit Engelhardt, Sebastian Fischetti, Ahmed Almheiri, Gary Horowitz, Don Marolf,William Kelly, and Mudassir Moosa for helpful discussions. This work is supported in part by theNational Science Foundation under Grant No. PHY12-05500. NE is also supported by the NationalScience Foundation Graduate Research Fellowship Program under Grant No. DGE-1144085. AWis also supported by the Simons Foundation.
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