Causality & holographic entanglement entropy

Consider a local quantum field theory (QFT) on a $d$-dimensional globally hyperbolic spacetime ${\cal B}$. The state on a given Cauchy slice⁴⁴4 Throughout this paper we will require all Cauchy slices to be acausal (no two points are connected by a causal curve). This is slightly different from the standard definition in the general-relativity literature, in which a Cauchy slice is merely required to be achronal. The reason is to ensure that different points represent independent degrees of freedom, which is useful when we decompose the Hilbert space according to subsets of the Cauchy slice. $\Sigma$ is described by a density matrix $\rho_{\Sigma}$; this could be a pure or mixed state. We are interested in the entanglement between the degrees of freedom in a region⁵⁵5 Technically, ${\cal A}$ is defined as the interior of a codimension-zero submanifold-with-boundary in $\Sigma$, $\partial{\cal A}$ is the boundary of that submanifold, and ${\cal A}^{c}:=\Sigma\setminus({\cal A}\cup\partial{\cal A})$. ${\cal A}\subset\Sigma$ and its complement ${\cal A}^{c}$. Following established terminology, we call the boundary $\partial{\cal A}$ the entangling surface.

The entanglement entropy is defined by first decomposing the Hilbert space ${\cal H}$ of the QFT into ${\cal H}_{{\cal A}}\otimes{\cal H}_{{\cal A}^{c}}$, after imposing some suitable cutoff.⁶⁶6 In the case of gauge fields, this decomposition is not possible even on the lattice. Instead, one must extend the Hilbert spaces ${\cal H}_{{\cal A}}$, ${\cal H}_{{\cal A}^{c}}$ to each include degrees of freedom on $\partial{\cal A}$, so that ${\cal H}\subset{\cal H}_{{\cal A}}\otimes{\cal H}_{{\cal A}^{c}}$ Buividovich:2008gq ; Donnelly:2011hn ; Casini:2013rba ; Donnelly:2014gva . The reduced density matrix $\rho_{{\cal A}}:=\operatorname{Tr}_{{\cal H}_{{\cal A}^{c}}}\rho_{\Sigma}$ captures the entanglement between ${\cal A}$ and ${\cal A}^{c}$; in particular, the entanglement entropy is given by its von Neumann entropy: $S_{{\cal A}}:=-\operatorname{Tr}\left(\rho_{{\cal A}}\ln\rho_{{\cal A}}\right)$. For holographic theories, we expect that this quantity has good properties in the large-$N$ limit,⁷⁷7 Technically, by “large-$N$” we mean large $c_{\text{eff}}$, where $c_{\text{eff}}$ is a general count of the degrees of freedom (see Marolf:2013ioa for the general definition of $c_{\text{eff}}$). unlike the Rényi entropies $S_{n,{\cal A}}:=-\frac{1}{n-1}\ln\operatorname{Tr}\left(\rho_{{\cal A}}^{n}\right)$ Headrick:2010zt ; Headrick:2013zda . Note that both quantities are determined by the eigenvalues of ${\rho_{{\cal A}}}$, and are thus insensitive to unitary transformations of ${\rho_{{\cal A}}}$.

Now, since $\Sigma$ is a Cauchy slice, the future (past) evolution of initial data on it allows us to reconstruct the state of the QFT on the entirety of ${\cal B}$. In other words, the past and future domains of dependence of $\Sigma$ , $D^{\pm}[\Sigma]$, together make up the background spacetime on which the QFT lives, i.e., $D^{+}[\Sigma]\cup D^{-}[\Sigma]={\cal B}$. Likewise, the domain of dependence of ${\cal A}$, $D[{\cal A}]=D^{+}[{\cal A}]\cup D^{-}[{\cal A}]$, is the region where the reduced density matrix ${\rho_{{\cal A}}}$ can be uniquely evolved once we know the Hamiltonian acting on the reduced system in ${\cal A}$.⁸⁸8 We remind the reader that $D[{\cal A}]$ is defined as the set of points in ${\cal B}$ through which every inextendible causal curve intersects ${\cal A}$. Note that, given that we have defined ${\cal A}$ as an open subset of $\Sigma$, $D[{\cal A}]$ is open subset of ${\cal B}$.

${\cal A}^{c}$ similarly has its domain of dependence $D[{\cal A}^{c}]$. However, unless ${\cal A}$ comprises the entire Cauchy slice, the two domains do not make up the full spacetime, $D[{\cal A}]\cup D[{\cal A}^{c}]\neq{\cal B}$, since we have to account for the regions which can be influenced by the entangling surface $\partial{\cal A}$. Denoting the causal future (past) of a point $p\in{\cal B}$ by $J^{\pm}(p)$ we find that we have to keep track of the regions $J^{\pm}[\partial{\cal A}]$ which are not contained in either $D[{\cal A}]$ or $D[{\cal A}^{c}]$. As a result, the full spacetime ${\cal B}$ decomposes into four causally-defined regions: the domains of dependence of the region and its complement, and the causal future and past of the entangling surface:

These four regions are non-overlapping (except that $J^{\pm}[\partial{\cal A}]$ both include $\partial{\cal A}$). See Fig. 2 for an illustration of this decomposition. Although this decomposition is fairly obvious pictorially, for completeness we provide a proof in §4 (cf. theorem 12).

The decomposition (1) is particularly convenient for formulating the QFT causality constraint. Recall that the eigenvalues of the reduced density matrix ${\rho_{{\cal A}}}$, and hence the Rényi and von Neumann entropies, are invariant under unitary transformations which act on ${\cal H}_{{\cal A}}$ alone or on ${\cal H}_{{\cal A}^{c}}$ alone. These include perturbations of the Hamiltonian and local unitary transformations supported in the domains $D[{\cal A}]$ or $D[{\cal A}^{c}]$. In particular, if we consider another region ${\cal A}^{\prime}$ of a Cauchy slice $\Sigma^{\prime}$ such that $D[{\cal A}]=D[{\cal A}^{c}]$ (as indicated in Fig. 2), then the state $\rho_{\Sigma^{\prime}}$ is related by a unitary transformation to the state $\rho_{\Sigma}$. It is clear that such a transformation can be constructed from operators localized in ${\cal A}$, and so does not change the entanglement spectrum of ${\rho_{{\cal A}}}$. Furthermore, if we fix the state at $t\to-\infty$, 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 $J^{-}[\partial{\cal A}]$, because otherwise we can imagine evaluating $S_{\cal A}$ by using a sufficiently early Cauchy slice $\Sigma^{\prime}\supset\partial{\cal A}$ that passes to the past of $R$. Similarly, if we fix the state at $t\to+\infty$, the spectrum can be affected only by perturbations in $J^{+}[\partial{\cal A}]$. In summary, we have the following properties of ${\rho_{{\cal A}}}$:

• •

  The entanglement spectrum of ${\rho_{{\cal A}}}$ depends only on the domain $D[{\cal A}]$ and not on the particular choice of Cauchy slice $\Sigma$. The spectrum is thus a so-called “wedge observable” (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 spectrum of ${\rho_{{\cal A}}}$ is insensitive to any local deformations of the Hamiltonian in $D[{\cal A}]$ or $D[{\cal A}^{c}]$.

These are the crucial causality requirements that entanglement (Rényi) entropies are required to satisfy in any relativistic QFT.

The essential result of this paper is that the HRT proposal for computing $S_{{\cal A}}$ satisfies these causality constraints. In the conclusions we will revisit the question of what the dual of ${\rho_{{\cal A}}}$, and thus of the data in $D[{\cal A}]$, might be.

Let us now restrict attention to the class of holographic QFTs, which are theories dual to classical dynamics in some bulk asymptotically AdS spacetime. To be precise, we only consider strongly coupled QFTs in which the classical gravitational dynamics truncates to that of Einstein gravity, possibly coupled to matter which we will assume satisfies the null energy condition.

The dynamics of the QFT on ${\cal B}$ is described by classical gravitational dynamics on a bulk asymptotically locally AdS spacetime ${\cal M}$ with conformal boundary ${\cal B}$, the spacetime where the field theory lives. We define $\tilde{\cal M}:={\cal M}\cup{\cal B}$. $\tilde{\cal M}$ is endowed with a metric $\tilde{g}_{ab}$ which is related by a Weyl transformation to the physical metric $g_{ab}$ on ${\cal M}$, $\tilde{g}_{ab}=\Omega^{2}g_{ab}$, where $\Omega\to 0$ on ${\cal B}$.⁹⁹9 These are necessary but not sufficient conditions for the spacetime to be asymptotically AdS. Causal domains on $\tilde{\cal M}$ will be denoted with a tilde to distinguish them from their boundary counterparts, e.g., ${\tilde{J}}^{\pm}(p)$ will denote the causal future and past of a point $p$ in $\tilde{\cal M}$ and ${\tilde{D}}[R]$ will denote the domain of dependence of some set $R\subset\tilde{\cal M}$.

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 $\asymp$, i.e.

Moreover, to denote regions that are spacelike separated from a point, we will use ${\cal S}(p)$ and ${\tilde{\cal S}}(p)$ in the boundary and bulk respectively,

Just as for other causal sets, we can extend these definitions to any region $R$, namely ${\cal S}[R]:=\cap_{p\in R}{\cal S}(p)$ is the set of points which are causally disconnected from the entire region $R$, etc.

Having established our notation for general causal relations, let us now specify the notation relevant for holographic entanglement entropy. As before we will fix a region ${\cal A}$ on the boundary. The HRT proposal Hubeny:2007xt states that the entanglement entropy $S_{\cal A}$ is holographically computed by the area of a bulk codimension-two extremal surface ${\cal E}_{\cal A}$ that is anchored on $\partial{\cal A}$; specifically,

In the static (RT) case, it is known that the extremal surface is required to be homologous to ${\cal A}$, meaning that there exists a bulk region ${\cal R}_{{\cal A}}$ such that $\partial{\cal R}_{{\cal A}}={\cal A}\cup{\cal E}_{\cal A}$. So far, it has not been entirely clear what the correct covariant generalization of this condition is. In particular, should it merely be a topological condition, or should one impose geometrical or causal requirements on ${\cal R}_{{\cal A}}$, for example, that it be spacelike? (A critical discussion of the issues involved can be found in Hubeny:2013gta .) In this paper, we will show that a clean picture, consistent with all aspects of field-theory causality, is obtained by requiring that ${\cal R}_{{\cal A}}$ be a region of a bulk Cauchy slice.¹⁰¹⁰10 Technically, similarly to ${\cal A}$, we define ${\cal R}_{{\cal A}}$ to be the interior of a codimension-zero submanifold-with-boundary of a Cauchy slice $\tilde{\Sigma}$ of $\tilde{\cal M}$ (with $\tilde{\Sigma}\cap{\cal B}=\Sigma$). Since $\tilde{\Sigma}$ itself has a boundary (namely its intersection with ${\cal B}$), the interior of a subset (in the sense of point-set topology) includes the part of its boundary along ${\cal B}$. Thus, ${\cal R}_{{\cal A}}$ includes ${\cal A}$ (but not ${\cal E}_{\cal A}$). We will call this the “spacelike homology” condition.¹¹¹¹11 If there are multiple extremal surfaces obeying the spacelike homology condition, then we are to pick the one with smallest area. However, in this paper we will not use this additional minimality requirement; all our theorems apply to any spacelike-homologous extremal surface.

The homology surface ${\cal R}_{{\cal A}}$ naturally leads us to the key construct pertaining to entanglement entropy, which we call the entanglement wedge of ${\cal A}$, denoted by¹²¹²12 While we have associated it notationally with the region ${\cal A}$, it depends only on $D[{\cal A}]$. ${\cal W}_{\cal E}[{\cal A}]$. This can be defined as a causal set, namely the bulk domain of dependence of ${\cal R}_{{\cal A}}$,

Note that the entanglement wedge is a bulk codimension-zero spacetime region, which can be equivalently identified with the region defined by the set of bulk points which are spacelike-separated from ${\cal E}_{\cal A}$ and connected to $D[{\cal A}]$. The latter definition has the advantage of absolving us of having to specify an arbitrary homology surface ${\cal R}_{{\cal A}}$ rather than just ${\cal E}_{\cal A}$ and $D[{\cal A}]$. As we shall see below, the bulk spacetime can be naturally decomposed into four regions analogously to the boundary decomposition (1); the entanglement wedge is then the region associated with (and ending on) $D[{\cal A}]$.

While we have focused on the regions in the bulk which enter the holographic entanglement entropy constructions, we pause here to note two other causal constructs that can be naturally associated with ${\cal A}$. First of all we have the causal wedge ${\cal W}_{\cal C}[{\cal A}]$ which is set of all bulk points which can both send signals to and receive signals from boundary points contained in $D[{\cal A}]$, i.e.,¹³¹³13 Following Hubeny:2012wa , we can also define a particular bulk codimension-two surface ${\Xi_{\cal A}}$, the causal information surface, to be the rim of the causal wedge; in fact, it is the minimal area codimension-two surface lying on $\partial{\cal W}_{\cal C}[{\cal A}]$.

(The entanglement wedge ${\cal W}_{\cal E}[{\cal A}]$ and causal wedge ${\cal W}_{\cal C}[{\cal A}]$ are in fact special cases of the “rim wedge” and “strip wedge” introduced recently in Hubeny:2014qwa as bulk regions associated with residual entropy.)

The second bulk causal domain which will play a major role in our discussion below is a region we call the causal shadow ${\cal Q}_{\partial{\cal A}}$ associated with the entangling surface $\partial{\cal A}$. We define this region as the set of points in the bulk ${\cal M}$ that are spacelike-related to both $D[{\cal A}]$ and $D[{\cal A}^{c}]$, i.e.,

For a generic region ${\cal A}$ in a generic asymptotically AdS spacetime, the causal shadow is a codimension-zero spacetime region; see Fig. 3 for an illustrative example.¹⁴¹⁴14 The bulk metric used in the plot for Fig. 3 is $$ds^{2}=\frac{1}{\cos^{2}\rho}\left(-f(\rho)\,dt^{2}+\frac{d\rho^{2}}{f(\rho)}+\sin^{2}\rho\,d\varphi^{2}\right),\qquad f(\rho)=1-\frac{1}{2}\,\sin^{2}(2\,\rho)\,.$$ The matter supporting this geometry satisfies the null energy condition as can be checked explicitly. In certain special (but familiar) situations, such as spherically symmetric regions in pure AdS (where ${\rho_{{\cal A}}}$ is unitarily equivalent to a thermal density matrix), it can degenerate to a codimension-two surface. In such special cases, the entanglement wedge and the causal wedge coincide Hubeny:2012wa . In general, the causal information surface for ${\cal A}$ and that for ${\cal A}^{c}$ comprise the edges of the causal shadow. For a generic pure state these causal information surfaces each recede from ${\cal E}_{\cal A}$ towards their respective boundary region but approach each other near the AdS boundary. Hence the geometrical structure of ${\cal Q}_{\partial{\cal A}}$, described in language of a three-dimensional bulk, is a “tube” (connecting the two components of $\partial{\cal A}$) with a diamond cross-section, which shrinks to a point where the tube meets the AdS boundary at $\partial{\cal A}$.

For topologically trivial deformations of AdS, in the absence of ${\cal E}_{\cal A}$ (i.e.  when the state is pure and ${\cal A}=\Sigma$) the causal shadow disappears, but intriguingly, even when ${\cal A}$ is the entire boundary Cauchy slice, the causal shadow can be nontrivial. This occurs for example in the AdS₃-geon spacetimes¹⁵¹⁵15 Since these describe pure states, the presence of a causal shadow region does not necessarily guarantee the presence of an extremal surface whose area gives the entanglement entropy contained within it. However, there will be some extremal surface spanning this region. Balasubramanian:2014hda and in perturbations of the eternal AdS black hole, such as those studied by Shenker:2013pqa . In such a situation we simply define the casual shadow of the entire boundary (dropping the subscript) as

Here ${\cal B}$ is understood generally to include multiple disconnected components; the causal shadow is the region spacelike separated from points on all the boundaries.

Having developed the various causal concepts which we require, let us now ask what the constraints of field-theory causality concerning entanglement entropy translate to in the bulk. The first constraint is that $S_{\cal A}$ should be a wedge observable, i.e. if $D[{\cal A}]=D[{\cal A}^{\prime}]$ then $S_{\cal A}=S_{{\cal A}^{\prime}}$. For this to hold in general, we need ${\cal E}_{\cal A}={\cal E}_{{\cal A}^{\prime}}$. The second concerns perturbations of the field-theory Hamiltonian. Such perturbations will source perturbations of the bulk fields, including the metric, that will travel causally with respect to the background metric. In particular, disturbances originating in $D[{\cal A}]$ will be dual to bulk modes propagating in ${\tilde{J}}^{+}\big{[}D[{\cal A}]\big{]}$ (if we fix the state in the far past) or in ${\tilde{J}}^{-}\big{[}D[{\cal A}]\big{]}$ (if we fix the state in the far future). If either of these bulk regions intersected ${\cal E}_{\cal A}$, the dual of local operator insertions in $D[{\cal A}]$ could change the area of ${\cal E}_{\cal A}$, meaning that the HRT proposal would be inconsistent with causality in the QFT. By the same token, the extremal surface cannot intersect ${\tilde{J}}^{+}\big{[}D[{\cal A}^{c}]\big{]}$ or ${\tilde{J}}^{-}\big{[}D[{\cal A}^{c}]\big{]}$. Since the region complement to union of the causal sets ${\tilde{J}}^{\pm}[D[{\cal A}]],{\tilde{J}}^{\pm}[D[{\cal A}^{c}]]$ is the set of points that are spacelike related to $D[{\cal A}]\cup D[{\cal A}^{c}]$, we learn that

In others words, using (7) we can say that ${\cal E}_{\cal A}$ has to lie in the causal shadow of $\partial{\cal A}$

It is known, based on properties of extremal surfaces, that ${\cal E}_{\cal A}$ lies outside the causal wedges ${\cal W}_{\cal C}[{\cal A}]$ and ${\cal W}_{\cal C}[{{\cal A}^{c}}]$ Hubeny:2012wa ; Wall:2012uf ; Hubeny:2013gba . This leaves open the possibility that the surface could still lie in the causal future (or past) of the boundary domain of dependence of ${\cal A}$ or ${\cal A}^{c}$. A particular worry arises in explicit examples in Vaidya-AdS geometries where the extremal surface lies on the boundary of ${\tilde{J}}^{+}\big{[}D[{\cal A}]\big{]}$. This then leaves open the question whether one might indeed be able to push ${\cal E}_{\cal A}$ into a causally forbidden region, by introducing appropriate deformations in $D[{\cal A}]$. A theorem of Wall Wall:2012uf (Theorem 6 of the reference), guarantees that this does not occur (modulo some assumptions).

We will prove an essentially equivalent statement in §4, directly for extremal surfaces in an asymptotically AdS spacetime. The main result however can be stated in terms of three simple causal relations:

In other words, the causal split of the bulk into spacelike- and timelike-separated regions from ${\cal E}_{\cal A}$ restricts to the boundary at precisely the boundary split (1). Given the decomposition (1), these causal relations imply that perturbations in $D[{\cal A}]\cup D[{\cal A}^{c}]$ are not in causal contact with ${\cal E}_{\cal A}$. So, as required, the extremal surface lies in the causal shadow.

As a consequence of this theorem, we will also show that, if there is a spacelike region ${\cal A}^{\prime}$ such that $D[{\cal A}^{\prime}]=D[{\cal A}]$, then there is a bulk region ${\cal R}_{{\cal A}^{\prime}}$ such that $\partial{\cal R}_{{\cal A}^{\prime}}={\cal A}^{\prime}\cup{\cal E}_{\cal A}$, so ${\cal E}_{\cal A}$ is spacelike-homologous to ${\cal A}^{\prime}$. Thus, the HRT formula gives the same entanglement entropy for ${\cal A}^{\prime}$ and ${\cal A}$, as required on the field-theory side.

A striking consequence of the theorems discussed above emerges when we consider spacetimes with two boundary components, and let ${\cal A}$ be (a Cauchy slice for) all of one component.

As a starting point, consider the eternal Schwarzschild-AdS_d+1 black hole in the Hartle-Hawking state, with a Penrose diagram shown in Fig. 4(a) below. The left and right boundaries of the diagram each have the topology ${\bf S}^{d-1}\times{\mathbb{R}}$. This geometry is believed to be dual to the CFT on the product spatial geometry ${\bf S}^{d-1}_{L}\times{\bf S}^{d-1}_{R}$, in the entangled “thermofield double” state Horowitz:1998xk ; Balasubramanian:1998de ; CarneirodaCunha:2001jf ; Maldacena:2001kr :

where $\mid\!E_{i}\rangle_{R,L}$ is the energy eigenstate of the CFT on ${\bf S}^{d-1}_{R,L}$.

Let $\Sigma_{R}$ lie on the $t=0$ slice of the right boundary, and consider the reduced density matrix for some region ${\cal A}\subset\Sigma_{R}$. Since this is a static geometry, its entanglement entropy $S_{\cal A}$ is computed by a minimal surface ${\cal E}_{\cal A}$ which never penetrates past the bifurcation surface ${\cal X}$ of the black hole Hubeny:2012ry .¹⁶¹⁶16 Note that the extremal surface does not come arbitrarily close to the horizon—it either includes a component that wraps the horizon, or stays a finite distance away from it Hubeny:2013gta . If we let ${\cal A}$ be the full Cauchy slice of one of the boundaries, say ${\cal A}=\Sigma_{R}$, the extremal surface precisely coincides with the black hole bifurcation surface, as indicated in Fig. 4. Note that ${\cal E}_{\cal A}$ lies on the edge of the causally acceptable region since ${\cal X}$ sits at the boundary of both ${\cal W}_{\cal C}[{\cal A}]$ and ${\cal W}_{\cal C}[{{\cal A}^{c}}]$, and therefore constitutes the entire causal shadow for this special case.

One might now wonder what happens if we deform the state (12). This is not an innocuous question. In time-dependent geometries, the global (teleological) nature of the event horizon implies that extremal surfaces anchored on the boundary can pass through this horizon Hubeny:2002dg . Furthermore, as first explicitly shown in AbajoArrastia:2010yt , even apparent horizons do not form a barrier to the extremal surfaces. Hence we see that, a priori, in a state which is a deformation of (12), ${\cal E}_{\cal A}$ is in danger of entering ${\cal W}_{\cal C}[{{\cal A}^{c}}]$.

The theorems we have stated above indicate that this does not happen. The question is, how precisely does the extremal surface ${\cal E}_{\cal A}$ avoid doing so? As a first step to answering this, consider a deformation of the static eternal case localized along a null shell emitted from the right boundary at some time. The corresponding metric is given by the global Vaidya-SAdS geometry, where both the initial (prior to the shell) and final (after the shell) spacetime regions describe a black hole.

Fig. 4b presents a sketch of the Penrose diagram of such a geometry, contrasted with the standard static eternal Schwarzschild-AdS black hole (Fig. 4a). The diagonal brown line represents the shell 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 (future and past, left and right) event horizons. The solid parts of these lines indicate where these event horizons coincide with apparent horizons (as well as isolated horizons); the dashed parts are parts of the event horizon which are not apparent horizons.

In such a geometry, let us again consider ${\cal A}=\Sigma_{R}$. Then our theorems guarantee that the extremal surface must lie on the null sheet separating regions $R_{c}$ and $P_{c}$: it is again spacelike-separated from both $D[\Sigma_{L}]$ and $D[\Sigma_{R}]$. (In fact, since the spacetime prior to the shell is identical to the eternal static case, the extremal surface remains in the 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 undeformed case. Indeed, any perturbation to Schwarzschild-AdS which emanates from (or reaches to) the right boundary cannot change the location of the original extremal surface by causality; it could at most generate a new extremal surface.

A less marginal case occurs when we symmetrically perturb both copies of the CFT as above. Consider a perturbation at $t=0$ such that spherically symmetric null shells are emitted both to the past and future on both sides of the diagram. One then obtains the Penrose diagram shown in Fig. 5; this has time-reflection symmetry about $t=0$, symmetry under exchanging the left and right sides, and the $SO(d)$ rotational symmetry.

According to the theorems above, the extremal surface must be spacelike-separated from both boundaries, when we take ${\cal A}=\Sigma_{R}$. Using both time and space reflection symmetry, it is clear that ${\cal E}_{\cal A}$ must sit in the center of the causal shadow ${\cal Q}$ of the two boundaries, spacelike separated from both.

In the general case of spherically symmetric spacetime (even in the absence of time or space reflection symmetry) there is an easy proof of our claim that ${\cal E}_{\cal A}$ must lie in the causal shadow. We proceed by contradiction: suppose that a spherical extremal surface ${\cal E}_{\cal A}$ lies in ${\tilde{J}}^{+}\left[\Sigma_{L}\right]$. This means that on a Penrose diagram, it lies somewhere in the top-left region; say it is the surface ${\cal F}_{\cal A}$ indicated in Fig. 5 (which by rotational symmetry is a copy of ${\bf S}^{d-1}$). Let us then consider the past congruence of null normal geodesics from ${\cal F}_{\cal A}$ towards ${\cal B}_{L}$. Since we assume that ${\cal F}_{\cal A}$ candidate surface lies in ${\tilde{J}}^{+}\left[\Sigma_{L}\right]$, past-going null congruences from the surface intersect ${\cal B}_{L}$ on a spacelike codimension-one surface. In other words, the area of the spheres grows without bound along this past-directed congruence.

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 also follows that the area along the congruence is guaranteed not to grow. Nor can the area go to zero along the congruence, since the area of the $S^{d-1}$ represented by each point on the Penrose diagram is finite. It therefore follows that our assumption about ${\cal E}_{\cal A}$ penetrating ${\tilde{J}}^{+}\left[\Sigma_{L}\right]$ must be erroneous; ${\cal F}_{\cal A}$ cannot be an extremal surface. Running a similar argument for the other unshaded regions in Fig. 5, we learn that the extremal surface must indeed lie in the causal shadow region, as denoted by the red surface ${\cal E}_{\cal A}$.

Indeed, in this particular case, the extremal surface lies at the point on the Penrose diagram where the future and past apparent horizons meet—the “apparent bifurcation surface”. The fact that it lies in the causal shadow is a consequence of the familiar fact that the apparent horizon can never be outside the event horizon, applied to both future 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 §4 will establish this in full generality.

In the next two sections we set out to prove the theorems stated in §2.3. The proof in our spherically symmetric case indicates that understanding null congruences leaving the extremal surface might play a key role. We will therefore spend some time in §3 examining null congruences emanating from bulk codimension-two surfaces in AdS₃, in order to develop a picture of the relevant causal domains, before embarking on a general proof in §4.

Since the $a=1$ surface is extremal, the null expansion $\Theta(\lambda;a=1)=0$ for each generator. For the surfaces with $a<1$, closer to the boundary, we expect that the expansion is positive and the congruence intersects the boundary in a spacelike curve inside $D[{\cal A}]=\{(t,x)\in{\mathbb{R}}^{1,1}\mid|t\pm x|\leq 1\}$. For curves with $a>1$, long ellipse, we expect the expansion to be negative. The resulting congruence should develop a caustic before reaching the boundary.

Due to the relative simplicity of the set-up, we can confirm these expectations explicitly. Since everything is time-symmetric, let us consider just the future-directed outgoing congruence:

Note that the endpoints of these generators at $\lambda=\infty$ are given by

A representative plot of the generators is given in Fig. 6 for $a=0.5$ (left) and $a=1.5$ (right). We see that when $a<1$, the generators don’t intersect each other before reaching the boundary, and they reach within $D^{+}[{\cal A}]$. On the other hand, when $a>1$, the generators intersect in a seam (drawn as thick blue curve, whose explicit expression is given below in (17)), before reaching the boundary (with the geodesic endpoints indicated by the red curves in Fig. 6). We call the points on this seam the cross-over points; non-neighbouring geodesics intersect at these points. This seam terminates in a caustic, which as always refers to the locus where neighbouring geodesics intersect.

We can determine the intersection between distinct geodesics in the bulk using the explicit expressions from (15). By symmetry of the set-up, we know that geodesics with opposite values of $x_{0}$ necessarily intersect, and they must do so at $x=x_{\times}=0$. Solving for the intersection of the pair of geodesics starting from $x_{0}$ and $-x_{0}$ we find that they meet at:

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 the same color). It is easy to see from (17) that the cross-over points terminate on the boundary at the future tip of $D^{+}[{\cal A}]$, i.e., at $z=0,\ x=0,\ t=1$, corresponding to the intersection of the boundary geodesics $x_{0}=\pm 1$. On the other hand, the cross-over seams for different $a$ start at the point in the bulk when neighbouring geodesics from $x_{0}\simeq 0$ intersect which happens at

To summarize, depending on whether $a$ is greater or less than 1, the congruence has qualitatively different behaviour, as illustrated in Fig. 7. For $a<1$ (depicted by colors from red toward green), the congruence reaches the boundary inside $D^{+}[{\cal A}]$, while for $a>1$, the generators intersect each other at the seam of crossover points (depicted by colors from red toward purple). At precisely $a=1$, all generators reach the boundary at the future tip of $D^{+}[{\cal A}]$, namely $z=0,\ x=0,\ t=1$.

Let us now analyze the expansion along this congruence. This can be calculated as the change in area along the wavefront

with

where $t^{\prime}(\lambda,x_{0})\equiv\frac{\partial}{\partial x_{0}}t(\lambda;x_{0})$ etc., using the expressions given in (15). While one can numerically solve for $\Theta(\lambda)$ it is easier to obtain the solution for small $\lambda$ and evolve using the Raychaudhuri equation.

Near $\lambda=0$, the leading order expression for $\Theta$ is:

This is plotted in the left panel of Fig. 8 (with same color-coding by $a$ as employed in Fig. 7). At the ends of the interval $x_{0}=\pm 1$, $\Theta_{0}$ vanishes (which is to be expected since the congruence approximates a larger one with $a=1$), while $\Theta_{0}$ reaches its extremum at the midpoint, $x_{0}=0$ (again, expected by symmetry), where $\Theta_{0}(x_{0}=0)=a\,(1-a^{2})$. Furthermore, $\Theta_{0}$ is positive for $a<1$ and negative for $a>1$; that is, the congruences are expanding for $a<1$ and converging for $a>1$). The former make it out to the boundary without intersecting, while the latter have a seam of cross-overs. As we will see below, the geodesics end in a curve of caustics, which touches the seam of cross-overs at the endpoint of the latter.

Given $\Theta_{0}$ as our initial condition, it is straightforward to solve the Raychaudhuri equation

to find the expansion along the geodesics. Here $\xi^{a}$ is the tangent vector to the null geodesics and $\sigma_{\mu\nu}$ is the shear of the congruence. For a one-dimensional congruence the shear trivially vanishes and the Ricci tensor contracted with null tangents likewise vanishes upon using the bulk equations of motion $R_{ab}=-2\,g_{ab}$, so (22) simplifies to:

Using (21), we find:

In Fig. 8 we have plotted this as a function of $\lambda$ for $x_{0}=0$, at which $\Theta=\frac{a\,(1-a^{2})}{1+a\,(1-a^{2})\,\lambda}$.

For $a>1$, we expect the congruence to develop a caustic where the expansion diverges. This occurs when infinitesimally nearby geodesics intersect each other. Eq. (24) shows that this can only occur for $a>1$, where the second term in the denominator is negative for positive $\lambda$. In this case $\Theta(\lambda)\to-\infty$ at a finite value of $\lambda=\lambda_{c}$,

for any $x_{0}$. The spacetime coordinates for the points along the congruence where this happens are given by

Viewed as a pair of parametric curves parametrized by $x_{0}$ which starts at $x_{0}=0$ and ends at $x_{0}=\pm 1$, the caustic seams are null curves, starting at the intersection point (18) and ending on the boundary at $z_{c}=0$, $x_{c}=\pm(1-a^{2})$, and $t_{c}=a^{2}$. Note that this is a finite distance on the boundary.

The divergence $\Theta\to-\infty$ signifies the presence of conjugate points, but their geometric meaning is a bit obscure in our discussion so far. The reason is as follows: as we see in Fig. 6 and can check explicitly, we generically have caustics in the neighbourhood of $x_{0}\simeq 0$, but more generally encounter cross-over points from the intersection geodesics symmetrically placed about $x_{0}=0$. The expansion is finite along the cross-over seam (17) for $x_{0}\neq 0$. This can be understood by realizing that the expansion is a local property of the nearby geodesics which doesn’t know about any other piece of the congruence. So nothing special ought to happen at the cross-over points which are non-local in the congruence, and indeed these are not conjugate points.

The clue as to the geometric meaning of $\Theta\to-\infty$ comes from plotting this locus on the surface of the null congruence (continued through the cross-over seam). This is presented in Fig. 9 by the thick red curves. We see that the surface intersects itself at the cross-over seam, beyond which the constant-$\lambda$ wavefronts form closed loops. On the sharp flank, these wavefronts turn around and locally become null; this is precisely where $A(\lambda,x_{0})$ vanishes and therefore $\Theta\to-\infty$.

The upshot of our calculations can be summarized as follows. Consider the null geodesic congruence emanating from a codimension-two spacelike surfaces ${\cal F}_{\cal A}\subset{\cal M}$ anchored on the boundary of a region ${\cal A}$ with $\partial{\cal A}={\cal F}_{\cal A}\cap{\cal B}$.

• •

  If ${\cal F}_{\cal A}\subset{\cal W}_{\cal C}[{\cal A}]$ then the congruence terminates inside $D[{\cal A}]$ along a spacelike boundary codimension-one surface.

• •

  If ${\cal F}_{\cal A}$ lies on the boundary of the causal wedge ${\cal W}_{\cal C}[{\cal A}]$ then the congruence intersects the boundary on the null surface $\partial D[{\cal A}]$.

• •

  If ${\cal F}_{\cal A}\subset{\tilde{\cal S}}\left[D[{\cal A}]\right]$ then the congruence finds itself terminated by a seam of cross-over points (and if continued further, would encounter caustic points prior to reaching the AdS boundary). The seam itself however reaches out to the boundary and ends on the future tip¹⁸¹⁸18 In higher-dimensional setting, $D[{\cal A}]$ itself may terminate in a crossover seam rather than a single point, which occurs when the null generators of $\partial D[{\cal A}]$ on the boundary themselves cross over. of $\partial D[{\cal A}]$.

This gives a clear picture of the causal domains for regions bounded by curves inside and outside of ${\cal W}_{\cal C}[{\cal A}]$. As we will see in our explicit proof, the extremal surface will in general lie outside of ${\cal W}_{\cal C}[{\cal A}]$; in special cases it can at best lie on the boundary, but never in the interior, of the causal wedge.

In this subsection we will describe our holographic setup and assumptions.¹⁹¹⁹19 We largely follow the setup and assumptions of section 3 of Gao:2000ga , with two exceptions: we remove the null generic condition and we add the condition that the boundary is totally geodesic for null geodesics (assumption (iii) below).

Let $({\cal M},g_{ab})$ be a connected spacetime, of dimension greater than or equal to 3, that can be embedded in a spacetime $({\bar{\cal M}},\tilde{g}_{ab})$, such that the boundary ${\cal B}$ of ${\cal M}$ in ${\bar{\cal M}}$ is a smooth timelike hypersurface in ${\bar{\cal M}}$, and such that $\tilde{g}_{ab}=\Omega^{2}g_{ab}$, where $\Omega$ is a smooth function on ${\bar{\cal M}}$ that vanishes on ${\cal B}$. (We do not assume that ${\cal B}$ is connected.) We define $\tilde{\cal M}:={\cal M}\cup{\cal B}$. On $\tilde{\cal M}$ we have a causal structure induced from $\tilde{g}_{ab}$, which in ${\cal M}$ agrees with that induced from $g_{ab}$. We make the following assumptions:

1. (i)

   $({\cal M},g_{ab})$ obeys the null energy condition.

2. (ii)

   $\tilde{\cal M}$ is globally hyperbolic.

3. (iii)

   Every null geodesic in $({\cal B},\tilde{g}_{ab})$ is a geodesic in $(\tilde{\cal M},\tilde{g}_{ab})$.²⁰²⁰20Assumption (iii) is equivalent to the following property of the extrinsic curvature $K_{ab}$ of ${\cal B}$ in $\tilde{\cal M}$: for any point $p\in{\cal B}$ and any null vector $k^{a}$ in the tangent space to ${\cal B}$ at $p$, $K_{ab}k^{a}k^{b}=0$. That it holds for an asymptotically AdS spacetime can be seen by working in Fefferman-Graham coordinates. If we set $\Omega=1/z$, where $z$ is the standard radial coordinate, then $K_{ab}=0$ (so all geodesics in ${\cal B}$ are geodesics in $\tilde{\cal M}$, i.e. ${\cal B}$ is totally geodesic). The property $K_{ab}=0$ is not preserved by Weyl transformations, and so does not hold for a general choice of $\Omega$, but the weaker condition $K_{ab}k^{a}k^{b}=0$ does (as can be seen either from a direct calculation or from the fact that the set of null geodesics is invariant under Weyl transformations).

We begin by showing that ${\cal B}$ is globally hyperbolic. We omit the proofs, which are very simple, cf., Galloway:1999bp . (For brevity, we will only indicate one time direction for each statement below, but the time-reversed statements are clearly equally valid.)

In this subsection, we will study null geodesics in $\tilde{\cal M}$. Assumption (iii) has the following useful implication:

Proof: Given a point $p$ in ${\cal B}$ and a non-zero null vector in the tangent space to ${\cal B}$ at $p$, there exists a null geodesic in ${\cal B}$ passing through $p$ with that tangent vector. By assumption (iii), it is a geodesic in $\tilde{\cal M}$, and by the uniqueness of geodesics it is the only one. Therefore no null geodesic passing through ${\cal M}$ can intersect ${\cal B}$ tangentially. Finally, since ${\cal B}$ is the boundary of $\tilde{\cal M}$ and is smooth, any smooth curve that intersects ${\cal B}$ at some point without ending there must be tangent to it. $\Box$

Now we constrain the behavior of congruences of null geodesics that pass through ${\cal M}$, using the fact that the metric $g_{ab}$ obeys the null energy condition and the fact that a geodesic that reaches ${\cal B}$ travels an infinite affine parameter.

Proof: We begin by working in the metric $g_{ab}$. Since the deviation vectors are everywhere spacelike, the expansion $\Theta$ is finite everywhere. On any geodesic that reaches ${\cal B}$, the affine parameter goes to infinity, so, by the null energy condition, $\Theta$ is nowhere negative, and therefore vanishes everywhere. Again using the null energy condition, the shear therefore vanishes everywhere also. Therefore, for any one-parameter family of geodesics that reach ${\cal B}$, the norm of the deviation vector $X^{a}$ is a positive constant along each geodesic.

We now return to $\tilde{\cal M}$, and switch to the metric $\tilde{g}_{ab}$. On ${\cal B}$, $X^{a}$ 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 choose the affine parameter $\lambda$ on each geodesic so that it intersects ${\cal B}$ at $\lambda=0$; hence, at $\lambda=0$, $X^{a}$ is tangent to ${\cal B}$. However, by lemma 5, $T^{a}$ is not tangent to ${\cal B}$. So $X^{a}=0$. Since this holds for every one-parameter family of geodesics, every connected set of geodesics that reach ${\cal B}$ intersects it at a point. $\Box$

As a warm-up for our main theorem of this subsection, we will now use lemma 6 to prove a version of the Gao-Wald theorem Gao:2000ga and a version of the topological censorship theorem Galloway:1999br .

Proof: Clearly $J^{+}(p)\subset{\tilde{J}}^{+}(p)\cap{\cal B}$. Let $t$ be a global time function on $\tilde{\cal M}$. Then if $t(q)<t(p)$ we have $q\notin{\tilde{J}}^{+}(p)$. Therefore, each connected component of ${\cal B}$ contains some points not in ${\tilde{J}}^{+}(p)$. Therefore, if ${\tilde{J}}^{+}(p)\cap{\cal B}\neq J^{+}(p)$, then $\partial{\tilde{J}}^{+}(p)\cap{\cal B}$ includes a hypersurface ${\cal S}$ in ${\cal B}$ that is not in $J^{+}(p)$. We will now show that ${\cal S}$ cannot exist.

$\partial{\tilde{J}}^{+}(p)$ consists of future-directed null geodesics starting at $p$ on which, except at the endpoints, every deviation vector is spacelike and orthogonal to the tangent vector. By lemma 5, each such geodesic either lies entirely in ${\cal B}$ or lies entirely in ${\cal M}$ except at its endpoints. In particular, the points in ${\cal S}$ must lie on geodesics that are entirely in ${\cal M}$ except at their endpoints. We thus consider the congruence of geodesics in ${\cal M}$ starting at $p$. Reversing its direction, every geodesic in this congruence reaches ${\cal B}$ (at $p$), so the expansion is nowhere negative. Therefore, in the forward direction, its expansion is nowhere positive. Thus the conditions of lemma 6 apply. Hence ${\cal S}$ consists of isolated points, contradicting the fact that it is a hypersurface in ${\cal B}$. $\Box$