A World without Pythons would be so Simple
AA World without Pythons would be so
Simple
Netta Engelhardt, Geoff Penington, , and Arvin Shahbazi-Moghaddam Center for Theoretical Physics, Massachusetts Institute of Technology,Cambridge, MA 02139, USA Center for Theoretical Physics and Department of Physics,University of California, Berkeley, CA 94720, U.S.A. and Institute for Advanced Study, 1 Einstein Dr, Princeton, NJ 08540, U.S.A. Stanford Institute for Theoretical Physics,Stanford University, Stanford, CA 94305 USA [email protected] , [email protected] , [email protected] Abstract:
We show that bulk operators lying between the outermost extremal surfaceand the asymptotic boundary admit a simple boundary reconstruction in the classicallimit. This is the converse of the Python’s lunch conjecture, which proposes that oper-ators with support between the minimal and outermost (quantum) extremal surfaces– e.g. the interior Hawking partners – are highly complex. Our procedure for recon-structing this “simple wedge” is based on the HKLL construction, but uses causal bulkpropagation of perturbed boundary conditions on Lorentzian timefolds to expand thecausal wedge as far as the outermost extremal surface. As a corollary, we establish theSimple Entropy proposal for the holographic dual of the area of a marginally trappedsurface as well as a similar holographic dual for the outermost extremal surface. Wefind that the simple wedge is dual to a particular coarse-grained CFT state, obtainedvia averaging over all possible Python’s lunches. An efficient quantum circuit convertsthis coarse-grained state into a “simple state” that is indistinguishable in finite timefrom a state with a local modular Hamiltonian. Under certain circumstances, the sim-ple state modular Hamiltonian generates an exactly local flow; we interpret this resultas a holographic dual of black hole uniqueness. a r X i v : . [ h e p - t h ] F e b ontents A uv and vi constraints 34B Alternative derivation of the vv constraint 34
Recent developments in the black hole information frontier have pointed to a holo-graphic geometrization of the degrees of freedom of the Hawking radiation [1–6]. Foran AdS black hole evaporating into a bath, the “entanglement wedge of the radiation”after the Page time includes a large part of the black hole interior, bounded by theminimal quantum extremal surface (QES) [7].This geometric description of the information naturally accounts for both the Pagecurve [8] and the Hayden-Preskill decoding criterion [9]. It also leads to a geometricalexplanation for the expectation of Harlow-Hayden [10] that decoding Hawking radiation– 1 – igure 1 . An illustration of the Python’s lunch. On the left, the geometry of a Cauchy slicefeaturing the titular python’s lunch between the two constrictions, the dominant QES X min and the non-minimal QES X , both of which lie behind the causal surface C . On the right, aspacetime diagram of the same. should be exponentially complex. Even though the interior degrees of freedom lie onthe radiation side of the minimal QES and so lie within the radiation entanglementwedge, they are still hidden behind a nonminimal QES; in the case of the single-sidedblack hole, the nonminimal QES is simply the empty set. The region between thenonminimal and minimal extremal surfaces was dubbed “the Python’s lunch” in [6],because appropriate Cauchy slices in the bulk (quantum) geometry have a constrictionat each extremal surface, together with a bulge in the middle (the eponymous “lunch”).See Fig. 1 for an illustration.The claim of [6] was that any bulk operator with support in the interior of aPython’s lunch should be exponentially difficult to decode, with an exponent that iscontrolled by the size of the bulge and grows as O (1 /G N ) in the semiclassical limit.The justification for this conjecture was based primarily on tensor network toy models,where the fastest known protocols for decoding operators inside a lunch use a Grover-search-based algorithm that takes exponential time.An additional important consistency check comes from the quantum focusing con-jecture [11], which is the quantum avatar of classical gravitational lensing when thenull energy condition is violated due to quantum corrections. Quantum focusing andglobal hyperbolicity ensure that no causal semiclassical Lorentzian evolution of thebulk geometry can result in causal communication from behind a Python’s lunch tothe asymptotic boundary, no matter what the asymptotic boundary conditions are.If such communication were possible, reconstruction of information from within thelunch could be implemented using only boundary time evolution with simple sourcesand the extrapolate dictionary relating bulk observables at the asymptotic boundaryto local boundary operators. From a boundary perspective, this is a very simple pro-cedure compared to the exponential complexity claimed to be necessary for operator– 2 –econstruction inside the lunch.Without input from nonperturbative quantum gravity (such as entanglement wedgereconstruction), simple reconstruction using only low-complexity, causally-propagatingoperators and sources is all that semiclassical Lorentzian gravity is capable of: i.e. allthat semiclassical gravity “sees”. Calculations and arguments that rely exclusively onsemiclassical gravity with no further input (e.g. Hawking’s original calculation) are thuscertainly restricted by the Python’s lunch proposal to recover no more than the domainof dependence between the outermost quantum extremal surface – the “appetizer” ofthe lunch – and the asymptotic boundary.So how much does purely semiclassical gravity actually recover? On the one hand, ifthe Python’s lunch conjecture is true, it is natural to expect that simple reconstructioncan in fact obtain the entire bulk up to the outermost extremal surface. This “converse”to the Python’s lunch conjecture is certainly true in tensor network toy models whereanything not in a lunch can be reconstructed using a simple unitary circuit. Any gapin gravitational theories between the simply reconstructible region and the start ofthe lunch would therefore be somewhat puzzling and demand explanation. On theother hand, simple reconstruction appears to be little more than a glorified versionof the HKLL procedure [12–14], which is supposed to recover just the so-called causalwedge: the region that can simultaneously send and receive signals from the asymptoticboundary. And generically the causal wedge and the outermost extremal wedge do notcoincide.To clarify this conundrum, let us first briefly review HKLL here, as it will be instru-mental for our work in this paper. The HKLL procedure is a reconstruction protocolfor bulk matter on a fixed background, in which the bulk fields (which can includegravitons [15], O (1 /N ) corrections [16, 17], interactions [18], and higher spins [16])are obtained via a non-standard Cauchy evolution from their boundary counterparts(related to them via the extrapolate dictionary). Quantitatively, φ ( x ) = (cid:90) dXK ( x ; X ) O ( X ) , (1.1)where K ( x ; X ) is a smearing function that depends on the spacetime geometry sup-ported on the set of boundary points spacelike-separated to x . The sense in whichHKLL is “simple” is evident: from a boundary perspective it consists of boundary Note that the validity of this non-standard “rotated” Cauchy problem is far from well-established(though see [19, 20] for proofs in certain cases). Our purpose here is not to put HKLL on a firmfooting, but rather to show that reconstruction of the simple wedge is as simple as HKLL. We shalltherefore assume HKLL, but any other simple reconstruction procedure for operators in the causalwedge would do just as well for our purposes. – 3 –ime evolution with local sources turned on. And local Hamiltonian evolution can besimulated efficiently using a quantum circuit.The immediate prediction therefore, as expressed in [21], is that HKLL can re-construct operators within the causal wedge. We might then expect that the simplyreconstructible region – which we shall henceforth refer to as the simple wedge – isto be identified with the causal wedge. Since the causal wedge is always a subset ofthe outermost quantum extremal wedge [7] and is generically a proper subset , thisleads to the undesirable no-man’s land between the simple wedge and the outermostquantum extremal surface.To see deeper into the bulk, we need to expand the causal wedge via the additionof simple boundary sources as proposed in [23, 24]. In the very special case when thegap between the outermost extremal surface and the causal wedge is Planckian, [25]showed that certain causal unitaries produce just enough backreaction to maximallyexpand the causal past or future. However, in generic spacetimes the gap region is non-empty even in the classical limit and can in fact be arbitrarily large. It was conjecturedin [23, 24] that it should be possible to fine-tune simple sources in order to “turn off”any extant focusing and so expand the causal wedge up to an apparent horizon, allwithout violating the null energy condition.A central result of this paper is an explicit, constructive derivation of this fact –in the limit where the bulk dynamics are classical and with a variety of matter fields.Furthermore, by evolving backwards and forwards in time using timefolds with differentboundary conditions, one can continue to iteratively expand the causal wedge, fromapparent horizon to apparent horizon, all the way to the outermost extremal surface.Combining this result with ordinary HKLL leads to simple reconstructions of arbitraryoperators in the outermost extremal wedge.It is easy to see how this works in the case of Jackiw-Teitelboim gravity [26, 27]minimally coupled to a (classical) massless free scalar. In this setup, the matter factor-izes into left and right movers; by changing the boundary conditions, we may “absorb”the right movers and turn off focusing on the future event horizon; this pushes thefuture event horizon backwards. We can then repeat the same procedure for the pasthorizon by evolving backwards in time; this will now push the past causal horizonbackwards. The shift will have likely revealed additional left-movers, so the procedureneeds to be iteratively repeated until it converges on a stationary bifurcate horizon. As shown in [22], quantum effects can allow the causal wedge to be outside the outermost extremalwedge when defined using time evolution couples the asymptotic boundary to an auxiliary system.However, this is only true if we do not include the auxiliary coupled system when defining the outermostextremal wedge. When comparing apples to apples by doing so, one indeed finds that the causal wedgeis still contained in the outermost extremal wedge. – 4 – C C C C C Figure 2 . A caricature of the procedure used to push the causal wedge towards the appe-tizer in JT gravity coupled to a classical massless scalar. The leftmost panel is the originalspacetime including left- and right-movers with reflecting boundary conditions. The causalsurface of the right boundary is C . In the middle panel, the left movers have been turnedoff, which causes the future event horizon to shift inwards. The resulting causal surface is C , which is null-separated from C . The final panel shows that the right movers have beenturned off, which causes the past event horizon to move inwards, shifting the causal surfaceto C . This shift reveals new left-movers in the causal wedge, which will have to be removedin subsequent zigzags along the past and future event horizons. This is illustrated in Fig. 2. The generalization to higher dimensions is significantlymore technically challenging – rather than removing sources of focusing entirely, it ismore practical to “stretch out” the focusing over the causal horizon and so dilute itseffect – but the essential intuition is the same.The original motivation in [23] for attempting to expand the causal wedge usingsimple sources was to understand the holographic dual of the simple entropy , definedas a maximization of the von Neumann entropy over all CFT density matrices with thesame one-point functions – with arbitrary time-ordered simple sources turned on aftersome initial time t – as the original CFT state. In other words, the simple entropycoarse-grains over all of the details of the state, except for simple observables that canbe measured in the future of the initial time t . It was conjectured in [23, 24] thatthe simple entropy is the boundary dual of the outer entropy , a bulk quantity thatcoarse-grains over the geometry behind the outermost apparent horizon null-separatedfrom the boundary at time t ; it is equal to (one quarter of) the area of the apparenthorizon. As a corollary of the results discussed above, we prove that this conjecture isindeed true whenever the bulk physics can be treated classically.What if we generalize the definition of the simple entropy to allow not just time-ordered insertions of simple operators, but insertions on arbitrary timefolds (and atarbitrary time)? In this case, there is no obstacle to seeing behind apparent horizons.– 5 – X X min C X C CPT
Figure 3 . The coarse-graining procedure of [23, 24] as applied to the outermost extremalsurface. The spacetime behind the outermost extremal surface X is discarded and replacedwith a CPT conjugate of the outermost extremal wedge. The rest of the spacetime is generatedby standard Cauchy evolution. By evolving the state backwards (and forwards) in time and then turning simple sources,it is possible to causally alter the spacetime near the apparent horizon, changing itslocation and “seeing” degrees of freedom that were originally hidden behind it. As perthe discussion above, the first obstruction that cannot be bypassed in this way is theoutermost extremal surface. Indeed, our results demonstrate that the simple entropywith arbitrary timefolds allowed is holographically dual to the area of the outermostextremal surface. Similarly, the density matrix ρ coarse whose von Neumann entropyis the simple entropy with timefolds allowed reconstructs exactly the entire outermostextremal wedge and no more. In fact, we can actually construct a complete spacetime inwhich the outermost extremal wedge is the entire entanglement wedge of one connectedasymptotic boundary obtained using the spacetime doubling procedure of [23, 24]; thus ρ coarse is the actual CFT state dual to the canonical purification as proposed in [23, 24]and proven in [28]. See Fig. 3.An immediate application of our result is then the construction of the CFT dual tothe simple wedge in the final spacetime where the causal and outermost extremal wedgescoincide. That is, this dual can be produced from ρ coarse via a set of simple operations,with the dual bulk result being a two-sided black hole in which the bifurcation surfaceis extremal. The significance of this statement is manifold: we prove that the causaland entanglement wedges coincide if and only if the CFT state has a local modularHamiltonian, which shows that finite time measurements cannot tell that the modularflow generated by the simple state is not local. In spacetimes with sufficient isotropy,the simple wedge CFT modular flow is in fact exactly local. This is analogous to atype of no-hair theorem: the set of holographic black holes with a stationary bifurcationsurface is identical to the highly limited set of states with local modular Hamiltoniansin the dual CFT.From the perspective of holographic complexity, we may therefore interpret the– 6 –bsence of a Python’s lunch in the dual theory as the CFT state being related by asimple circuit to a rather special state with local modular flow (or at least indistin-guishable from local in finite time). The world, it would seem, is rarely simple; pythonsare ubiquitous. An explicit example of how a python might spring on an unsuspectingholographer in what would prima facie appear to be a python-less spacetime will beprovided in our upcoming work [29].The paper is structured as follows. In Section 2, we define the outermost extremalwedge and the simple wedge, and we prove that the former is well-defined. In Section 3we showcase our procedure for the simple case of JT gravity coupled to a massless(classical) scalar. In Section 4, we prepare the perturbation that moves the causalhorizon backwards along a the future event horizon in higher dimensional gravity (witharbitrary, null energy condition-satisfying matter), and we prove that the requiredperturbation satisfies the constraint equations. Section 5 describes the zigzag portionof the procedure and completes the proof. We discuss the implications of our results,from the dual to the area of the outermost extremal surface to the nature of the simplestate, in Section 6. We finish with a discussion of generalizations and other implicationsin Section 7. Assumptions and Conventions:
The bulk spacetime (
M, g ) is assumed to be clas-sical with the dynamics governed by the Einstein field equation, i.e. we work in thelarge- N and large λ limit of AdS/CFT except where otherwise stated. We will assumethe AdS analogue of global hyperbolicity [30]. We also assume that the initial spacetimeunder consideration is one that satisfies the Null Energy Condition (NEC): T ab k a k b ≥ T ab is the stress energy tensor and k a is any null vector. We will demonstratethat our perturbations of the spacetime maintain the NEC. All other conventions areas in [31] unless otherwise stated. • We shall use J ± to refer to the bulk causal future and past and I ± to refer to thebulk chronological future and past. Given a closed achronal set S , we use D [ S ] todenote its domain of dependence, which we shall take to contain its boundary, asin [31]. D + [ S ] and D − [ S ] refer to the future and past components of the domainof dependence. • Hypersurfaces will refer to codimension-one embedded submanifolds of arbitrarysignature. – 7 –
By a “surface” we will always mean an achronal, codimension-two embeddedsubmanifold which is Cauchy-splitting [32]. Two surfaces σ and σ are homol-ogous whenever there exists a hypersurface H such that ∂H = σ ∪ σ . We willbe primarily interested in surfaces homologous to (partial) Cauchy slices of theasymptotic boundary (CFT (sub)regions). • Let Σ be a Cauchy slice containing a surface σ homologous to a boundary(sub)region R . By definition, σ splits Σ into two disjoint components that we willdenote Int Σ [ σ ] and Out Σ [ σ ], where the conformal completion of the latter con-tains the boundary subregion R . We define W σ ≡ D [Out Σ [ σ ]], the outer wedgeof σ . Similarly, we define I σ ≡ D [Int Σ [ σ ]], the inner wedge of σ . See also [24]. • For a smooth surface σ homologous to a boundary (sub)region, we denote by k a and (cid:96) a the unique future-directed orthogonal null vector fields on the C subsetsof σ pointing towards Out Σ [ σ ] and towards Int Σ [ σ ] respectively. • We define ∂ + W σ = ∂D + [Out Σ [ σ ]] and ∂ − W σ = ∂D − [Out Σ [ σ ]]. When σ issmooth, ∂ + W σ and ∂ − W σ can be constructed by firing null congruences start-ing from k a and (cid:96) a , terminating the congruence at caustics and non-local self-intersections [31, 33]. • Given any orthogonal null vector field k a on surface σ , θ ( k ) denotes the expansionof σ along k a . We will refer to the following types of σ based on its expansions: – A compact σ is trapped if θ ( k ) < θ ( (cid:96) ) <
0, and marginally trapped if θ ( (cid:96) ) < θ ( k ) = 0. – σ is extremal if θ ( k ) = 0 and θ ( (cid:96) ) = 0. By linearity, σ is then stationary underdeformations along any direction. • The future and past causal horizons associated to any boundary spacetime region R ⊂ I are defined as ∂J − [ R ] and ∂J + [ R ] respectively. By convention, we use k a and (cid:96) a to refer to the generators of the future and past horizons respectively. Morespecifically, the future and past event horizons are defined as H + ≡ ∂J − [ I ] and H − ≡ ∂J + [ I ]. The causal surface can be defined as C ≡ H + ∩ H − andthe causal wedge is W C ≡ J + [ I ] ∩ J − [ I ][34]. An important result in generalrelativity – which follows from NEC and cosmic censorship – is that future causal k a is not uniquely defined in places where σ is not C . However, the expansion still has a definitesign, which can be computed via a limiting procedure. Note that although the term causal wedge is common in the literature, unlike the entanglementwedge it does not refer to a domain of dependence. – 8 –orizons satisfy an area law: the areas of their cross sections do not decrease aswe move the cross section to the future [35, 36]. In particular, this means thatany congruence of null generators on a future horizon has nonnegative expansion.By time-reversal symmetry, a “reverse” area law holds for past horizons. • We define the terminated horizons H + C ≡ H + ∩ J − [ I ] and H − C ≡ H − ∩ J − [ I ].These are natural definition for us since we are interested in perturbations of H + and H − caused by causal boundary sources. Three bulk regions are under consideration here: the outermost extremal wedge , the causal wedge, and the simple wedge. We will ultimately argue that the outermostextremal wedge is in fact the simple wedge, but in order to avoid subscribing to ourown conclusions before we have demonstrated them, we introduce terminology thatdistinguishes between the two.We will argue for the equivalence between the outermost extremal and simplewedges by showing that simple operations and sources (together with a finite numberof time-folds) are sufficient to shift the causal wedge so that it comes arbitrarily closeto coinciding with the outermost extremal wedge. While our primary results are forcompact extremal surfaces, many of our intermediate results remain valid for boundary-anchored surfaces. In Sec. 7.1 we will discuss in more detail the extent to which ourresults apply to the latter case.We have already defined the more familiar causal wedge in the introduction. Letus now give a precise definition of the outermost extremal wedge and the simple wedge.Intuitively, the extremal wedge is defined as the analogue of the entanglementwedge for the outermost extremal surface – be it minimal or not. So before definingthe outermost extremal wedge, we must prove that a unique outermost extremal surfaceexists in the first place:
Proposition 1.
If there exists more than one extremal surface homologous to a con-nected component of I , then exactly one is outermost; i.e. there exists an extremalsurface X contained in the outer wedge W X (cid:48) of all other extremal surfaces X (cid:48) homolo-gous to I . We will prove this proposition using a series of three lemmas:
Lemma 1.
Given two surfaces σ and σ homologous to I , W σ ∩ W σ is a domain ofdependence. – 9 – roof. Let Σ be a Cauchy slice containing σ . We define a new surface Σ (cid:48) = ∂ (( J + [Σ] ∪ J + [ σ ]) ∩ I − [ σ ]), where I − [ σ ] ≡ M − I − [ σ ]. Since every inextensible timelike curve atsome point in its past is outside J + [Σ] ∪ J + [ σ ], but eventually ends up in J + [Σ] ∩ I − [ σ ],and since no timelike curve can exit ( J + [Σ] ∪ J + [ σ ]) ∩ I − [ σ ]) after entering it, Σ (cid:48) isCauchy. Note that, despite appearances, the definition of Σ (cid:48) is invariant under time-reversal symmetry.We will now show that W σ ∩ W σ is the domain of dependence of H = Σ (cid:48) ∩ W σ ∩ W σ . Any causal curve intersecting W σ ∩ W σ needs to intersect Σ ∩ W σ either (i) in W σ − ∂W σ or (ii) outside of it . In case (ii), the causal curve needs to leave J − [ σ ]in W σ after intersecting Σ or enter J + [ σ ] in W σ before intersecting Σ. Therefore, inboth cases i and ii we conclude that the causal curve intersects H .Let V now be a domain of dependence that intersects I . Then there must exist an“edge” surface σ homologous to I such that V = W σ . More precisely, σ can be definedas the set of points p ∈ ∂V such that in any small neighborhood of p any inextensibletimelike curve crossing p only intersects V at p . Lemma 2.
Let V = W σ ∩ W σ and let σ be the edge of V as defined above. Then, σ ⊂ σ ∪ σ ∪ ( ∂ + W σ ∩ ∂ − W σ ) ∪ ( ∂ − W σ ∩ ∂ + W σ ) .Proof. Clearly, the edge is contained in ∂V ⊂ ∂W σ ∪ ∂W σ . Say some point p in theedge of V were in ∂ + W σ − σ . Then, every timelike curve crossing p leaves W σ (andtherefore V ) in I + ( p ) and enters W σ in I − ( p ). Furthermore, for p to be in the edge of V it needs to be in W σ , but I − ( p ) must not intersect W σ . Therefore, p needs to bein ∂ − W σ . Together with the time reverse of this argument and also switching σ and σ , we conclude that the edge of V must be contained in σ ∪ σ ∪ ( ∂ + W σ ∩ ∂ − W σ ) ∪ ( ∂ − W σ ∩ ∂ + W σ ).Lastly, we will state a lemma from Sec. 2.2 of [37], without providing the proof(see also Appendix B of [6] for a similar discussion). The lemma assumes the existenceof a stable maximin surface [30] in any domain of dependence. Lemma 3. If σ is a surface homologous to I satisfying θ ( k ) ≤ and θ ( (cid:96) ) ≥ , thenthere exists an extremal surface Y ⊂ W σ homologous to I . Let us provide some intuition for Lemma 3. The restricted-maximin prescriptionreturns a surface in W σ which is homologous to I and is minimal on some Cauchy sliceof W σ . If σ satisfies θ ( k ) < θ ( (cid:96) ) >
0, then the maximin surface cannot intersect σ , since its area would get smaller under deformations away from such intersections.– 10 –n [37], it was further shown that the max property of the maximin surface prohibitsintersections with ∂W σ − σ . This shows that the maximin surface is in the interior of W σ and thus extremal. Furthermore, it was argued that even when the inequalities arenot strict, there still exists an extremal surface homologous to I in W σ even thoughin the surface might lie on ∂W σ .We are now ready to prove Proposition 1: Proof.
In this proof any “surface” will mean a surface that is homologous to I . Letan extremal surface X be called exposed if there does not exist any extremal surface Y (cid:54) = X such that Y ⊆ W X . Let us first argue that there must always exist at least oneexposed surface. Define a partial ordering on extremal surfaces by declaring X ≥ Y if and only if W X ⊆ W Y . Note that exposed surfaces would correspond to maximalelements with respect to this partial order, while an outermost extremal surface wouldbe a greatest element. Upper bounds exist for any chain because monotonicity andboundedness (there are no extremal surfaces near asymptotic infinity) ensure that anysequence { X n } of extremal surfaces with W X n +1 ⊆ W X n converges to an extremalsurface X ∞ with W X ∞ ⊆ W X n for any finite n . Hence, by Zorn’s lemma, at least onemaximal element, i.e. exposed surface, exists.Now suppose, by way of contradiction, that there exists an exposed surface X that is not outermost, i.e. there exists some other extremal surface X such that W X (cid:42) W X . Let V = W X ∩ W X (by definition then V ⊂ W X ). By Lemma 2 and3, V = W σ for some surface σ ⊆ X ∪ X ∪ ( ∂ + W X ∩ ∂ − W X ) ∪ ( ∂ − W X ∩ ∂ + W X ).Let us consider each component of the set to which σ belongs separately. The first twoare subsets of extremal surfaces and hence have zero expansion. By focusing, the nullhypersurface ∂ + W X and ∂ + W X are non-expanding towards the future, while the nullhypersurface ∂ − W X and ∂ − W X are non-expanding towards the past. Therefore, σ satisfies θ ( k ) ≤ θ ( (cid:96) ) ≥ , and hence by Lemma 3, there exists an extremal surface Y contained in W σ = W X ∩ W X . The existence of this surface means that X is notexposed, giving our desired contradiction.As an aside, note that extending the last part of this argument to quantum extremalsurfaces, under assumption of the quantum focusing conjecture requires a small amountof extra work because the definition of quantum expansion is nonlocal. However strong This σ may have kinks in which case the expansions are technically not well-defined. However, weexpect that all of our results in this section generalizes easily to such surfaces using standard geometrictechniques. Intuitively, by slightly smoothing out the kinks we can get a new infinitesimally nearbysurface with very large expansions of the desired signs around the kinks. – 11 –ubadditivity is enough to ensure that the quantum expansion of σ satisfies the desiredinequalities.It is now easy to define the outermost extremal wedge: Definition 1.
Let X be the outermost extremal surface for a connected component of I . The outermost extremal wedge W X is the outer wedge of X . Next we would like to define the simple wedge. Conceptually, the simple wedge isthe largest bulk region that can be reconstructed from the near boundary state of thebulk fields using exclusively the bulk equations of motion. Consider some boundarystate ρ whose bulk dual ( M, g ) we would like to reconstruct and evolve it to the farpast and future with some Hamiltonian. In the classical regime, following Eq. (1.1),HKLL prescribes the values of the bulk fields in W C from the set of one-point functionsof their corresponding local boundary operators on I . In fact, the bulk equations ofmotions are sufficient to reconstruct the maximal Cauchy development of W C – whichwe can denote by W C .Cosmic censorship in general [38, 39] and causal wedge inclusion in particular [30,40, 41] guarantees that the causal wedge contains no extremal surfaces in its interior: W C ⊆ W X . It is therefore impossible to reconstruct the region behind X causally.HKLL alone, however, appears to prima facie fail at an earlier stage: the non-standardCauchy evolution appears to stop short of recovering the gap region between W C and W X , which is generically non-empty.What if we evolve ρ using a different Hamiltonian? Consider turning on a set ofCFT operators at various times during the evolution. This would “extract” a newset of one-point functions from ρ and therefore has the potential to expand the recon-structible region. In keeping with our philosophy of the simple wedge, we must restrictto sources that have a (semi-)classical bulk dual. Therefore, following [23, 24], we referto boundary sources as simple if the bulk fields that they produce propagate causallyinto the bulk from the boundary insertion point – and restrict to such sources hence-forth. The change in time evolution when such simple sources are applied within someboundary time interval [ t i , t f ] is given by the following time-ordered operator: E = T exp (cid:20) − i (cid:90) t f t i dt (cid:48) J ( t (cid:48) ) O J ( t (cid:48) ) (cid:21) , (2.1)where J ( t ) is a simple source and O J is its corresponding simple operator. Note that O J might involve spatial integrals of local boundary operators O ( t (cid:48) , x (cid:48) ). An example ofa simple operator is a spatial integral of a single-trace operator of the boundary gaugetheory. – 12 –dding E to the evolution, say in a future-directed timefold, changes the spacetimefrom M to some M (cid:48) . By causality, M − J + [ t i ] ⊂ M (cid:48) . In particular, the perturbationto the spacetime is localized away from the past event horizon H − . However, sourceslike Eq. (2.1) will typically change where the new future event horizon H + intersects H − . In particular, suppose that we find simple sources that “expand” the causalwedge, i.e. place the new causal surface C (cid:48) in the future of C on H − . Said in theCFT language, the new set of one-point data reconstructs a W (cid:48) C that contains H − C .Furthermore, knowing the bulk equations of motion and the original Hamiltonian, wecan reconstruct the Cauchy development of H − C (cid:48) , a wedge in the original spacetimethat contains W C as a proper subset.It is natural to define the simple wedge according to the maximal success of thisprocedure: Definition 2.
The simple wedge is the maximal bulk region that can be reconstructedfrom simple operators acting on the dual CFT state, with the inclusion of simple sourcesand timefolds.
Although we have defined the simple wedge in the context of classical field theoryin the bulk, it is important to note that HKLL can reconstruct the quantum state ofthe bulk propagating in the causal wedge at each step. For bulk fields in the 1 /N expansion, Eq. (1.1) provides the dictionary between local bulk operators and simpleCFT operators, realizing this reconstruction in the Heisenberg picture.Finally, we close this section by relating the causal and outermost extremal wedges.Intuitively, the causal surface should coincide with the extremal surface if and only ifthere is no focusing whatsoever on the horizons. However, because the extremal wedgeis defined as a domain of dependence and the causal wedge is defined in terms of causalhorizons, it does not immediately follow that the two must coincide in the absence offocusing. To reassure the reader, we prove the following lemma: Lemma 4.
Let X be the outermost extremal surface homologous to one or more con-nected components of I . W X = W C if and only if X is a bifurcation of stationaryhorizons (and thus X = C ).Proof. If W C = W X , then ∂ W C = ∂W X . If both wedges had been defined in termsof domains of dependence, it would immediately follow that C = X . However, since W C is not defined as a domain of dependence, we have to work a little harder. Thecomponent of ∂ W C which is spacelike separated to every point in Int[ W C ] is identicalto the component of W X which is spacelike-separated to every point in Int[ W X ]. By Here t i stands for a timeslice of I geometrically. – 13 –efinition of the causal wedge, the former is C . Since W X is generated by the domainof dependence of a hypersurface H whose boundary in M is X , X is exactly the setof points that are spacelike separated from every point in Int[ W X ]. This immediatelyshows that C = X . Because θ ( n ) [ X ] = 0 for all n a in the normal bundle of X , we findthat θ ( (cid:96) ) [ C ] = 0 = θ ( k ) [ C ]. By the NCC, a future (past) causal horizon can only havevanishing expansion on a slice if it has vanishing expansion everywhere to the future(past) of that slice. So ∂ W C and subsequently ∂W X is generated by stationary horizons,and X is a bifurcate stationary horizon.If X is a bifurcate stationary horizon, then results of [32, 33] immediately imply that ∂W X is the union of two truncated stationary horizons H ± . Since H + ∩ ∂M = ∂D + , H + is a past-directed null congruence fired from i + . By the theorems of [32], ∂J − [ I ] = H + up to geodesic intersections. Since H + is stationary, it has no intersections, so ∂J − [ I ] = H + . Similarly ∂J + [ I ] = H − . Thus ∂W X = ∂ W C and W X = W C by thehomology constraint.Note that this result remains valid for quantum extremal surfaces assuming the quan-tum focusing conjecture and a suitable generalization of AdS hyperbolicity. Let us illustrate our iterative procedure for removing matter falling across the pastand future causal horizons in a simple toy model of JT gravity minimally coupledto a massless scalar field ϕ (with no direct coupling between the dilaton Φ and ϕ ).The absence of propagating degrees of freedom of the gravitational field as well asthe factorization of the bulk matter into left- and right-movers are simplifications thatnaturally do not generalize to higher dimensions; nevertheless the procedure itself iswell-illustrated in this setting, which we include for pedagogical reasons. The additionalcomplications introduced in higher dimensions are resolved in subsequent sections.Due to focusing resulting from the scalar field ϕ , the bifurcation surface will gener-ically not be extremal, i.e. ∂ n Φ | C (cid:54) = 0 (3.1)where n a is some vector normal to the causal surface C (in particular, for null n a future-outwards directed, this would be positive; similarly for a time-reverse). As aconsequence of the highly limited number of degrees of freedom in the problem, theextremality failure can only be a result of focusing: the future causal horizon willexperience focusing due to the ϕ left-moving modes and the past causal horizon willexperience focusing due to the ϕ right-moving modes. Our procedure instructs us to– 14 – A C B δv C lim f u t u r e e v e n t h o r i z o n p a s t e v e n t h o r i z o n Figure 4 . An illustration of the approach to the limit point C lim , where C A and C B areinfinitesimally close to the limiting point. first remove the source of focusing of the future horizon by modifying the boundaryconditions of ϕ , which we can easily do by implementing absorbing boundary conditionsfor the right movers in order to remove all the left-moving modes. This removes focusingfrom the future causal horizon, which pushes the horizon deeper into the bulk. As aconsequence, the new causal surface (which is now marginally trapped) – let us call it C – is pushed further along the past event horizon, which remains unmodified by thisprocedure. This first step is illustrated in the second panel of Fig. 2.To turn off focusing of the past horizon, we evolve backwards in time, imposingboundary conditions that remove the right-movers. The past event horizon movesbackwards, and the new causal surface C (which is now marginally anti-trapped) isdisplaced from C along the future event horizon of C . However C is not necessarilyextremal since shifting the past causal horizon reveals a part of the spacetime that waspreviously not included in the causal wedge: in particular, new left-moving modes cannow appear in the causal wedge. This is illustrated in the third panel of Fig. 2. Thispiecewise-null zigzag procedure thus shifts the causal surface deeper into the bulk; wemay simply repeat the zigzag iteratively.In classical gravity, the focusing theorem and cosmic censorship (or strong asymp-totic predictability) together guarantee that no extremal surface is ever in causal contactwith I : so the zigzag procedure can never modify an extremal surface nor move thecausal surface deeper than any extremal surface. Thus the outermost extremal surfaceis an upper bound on the success of the procedure. Our goal, of course, is to show thatthis upper bound is in fact attained.Because the success of this procedure is bounded by the outermost extremal sur-face and because furthermore the procedure moves the surface monotonically inwards,– 15 – limiting causal surface C lim exists, and the corresponding causal wedge does not in-tersect any extremal surface. We will now argue that C lim is in fact extremal. Let C A be a causal surface obtained via iterative zigzags which is infinitesimally close to C lim ;without loss of generality we may take C A to be in the marginally trapped portion ofthe zigzag (i.e. the left-movers had just been removed). Let v , u be the affine param-eters along the future and past event horizons, respectively, in the spacetime in which C A is the causal surface. By construction, C A has no expansion along the future eventhorizon: ∂∂v Φ | C A = 0 , (3.2)and because by assumption it is not identical to C lim , ∂∂u Φ | C A < . (3.3)Let C B be the causal surface obtained by removing the right-movers from the spacetimewhere C A is the causal surface. By construction ∂∂u Φ | C B = 0 . (3.4)By the zigzag procedure, C A and C B are null-separated along the “old” future eventhorizon: i.e. along the null congruence that is the future event horizon in the spacetimein which C A is the causal surface. Let δv be the amount of affine parameter separating C A and C B . See Fig. 4 for an illustration. Since the points C A and C B must beinfinitesimally close to one another (since both infinitesimally near C lim ), the spacetimemetric in that neighborhood may be approximated as locally flat instead of AdS ; using u and v as coordinates: ds = − dudv. (3.5)In these coordinates, it is trivial to relate δv to the change in ∂ u Φ along v . In particular,we may bound it from below: δv ≥ ∂ u Φ ∂ v ∂ u Φ | max , (3.6)where ∂ v ∂ u Φ | max is the maximum value of ∂ v ∂ u Φ on the δv interval. Similarly defining δu as the null separation between C B and the next causal surface after again removingleft movers, we obtain an analogous bound: δu ≥ ∂ v Φ ∂ u ∂ v Φ | max . (3.7)Under the assumption that ∂ u ∂ v Φ is bounded from above in this neighborhood (whichwe generically expect to be true), δv and δu approach zero no slower than ∂ u Φ and ∂ v Φ– 16 –pproach zero: thus C lim must be extremal. Because focusing arguments ensure thatthe causal wedge is always contained in the outermost extremal wedge, it must be theoutermost extremal surface. Our task in higher dimensional gravity is now clear: we must find a perturbation thatremoves focusing from the causal horizons (without violating the null energy conditionanywhere in the perturbed spacetime), thus shifting the causal wedge deeper in. Whatkind of perturbation δg would move the causal surface towards rather than away fromthe appetizer? On a heuristic level, we are looking to open up the bulk lightcones sothat more of the bulk is in causal contact with the asymptotic boundary. In search-ing for such a perturbation, we may build on the intuition of the boundary causalitycondition [42], which states that the inequality (cid:90) δg ab k a k b ≥ k a ) is equivalentto demanding that perturbations δg of pure AdS source focusing (as opposed to de-focusing). Here we are looking to do the opposite: we are looking to undo focusing,so it makes sense to look for a perturbation that satisfies an opposite inequality, with δg kk < H + C . It is a priori not clear that it is possible to find a pertur-bation that simultaneously satisfies this inequality and also results in a spacetime thatsolves the Einstein equation. To prove this point, we must show that the requisite δg solves the characteristic constraint equations on the event horizon.In this section, we will prove that as long as the causal surface is not marginallyouter trapped – i.e. as long as θ ( k ) [ C ] (cid:54) = 0, it is possible to find exactly such a perturba-tion that (1) satisfies the characteristic constraint equations on the causal horizon and(2) shifts the causal surface deeper into the bulk. The procedure is roughly as follows:we prescribe a δg deformation on H + ; some elements of this δg resemble the “leftstretch” construction [43, 44] involving a rescaling of the generators of certain achronalnull hypersurfaces – intuitively, this dilutes the infalling content on H + and in turnreduces focusing. We then demonstrate that the gravitational constraints on δg alongwith boundary conditions fix the requisite components of δg in such a way that theperturbed spacetime has a larger causal wedge. In Sec. 5, we will argue how repeatingthese perturbations pushes in the causal surface up to an apparent horizon in a giventimefold, and to the appetizer using several timefolds.– 17 – = v = u=v=0 u = v = Figure 5 . On the left: three-dimensional (left) illustration of the past ( v = 0) and future( u = 0) event horizons, with a slicing of the future event horizon given by the intersection of u = 0 with past causal horizons originating from complete slices of the I . On the right: aconformal diagram illustrating the same. We will call the generators of the future and past event horizons k a and (cid:96) a respec-tively. We will extend (cid:96) a to the entire spacetime by picking a smooth Cauchy foliation {C α } α of I and defining (cid:96) a to be the bulk generators of ∂I + [ C α ]. This defines a nullfoliation of J + [ I ] by past causal horizons; The past event horizon, which C lies on, isa member of this foliation.We adopt the coordinate and gauge choices of [24]: first, we introduce double nullcoordinates ( u, v ) , where k a = (cid:18) ∂∂v (cid:19) a ; (cid:96) a = (cid:18) ∂∂u (cid:19) a . (4.2)In these coordinates, the causal surface is at u = v = 0, the future event horizon H + is at u = 0 and the past event horizon H − is at v = 0. See Fig. 5. We can furtherfix the gauge in a neighborhood of u = 0 so that the metric there takes the form: ds = − dudv + g vv dv + 2 g vi dvdy i + g ij dy i dy j (4.3)where i, j denote the transverse direction on H + . At u = 0 exactly, we of courserequire that k a = ( ∂/∂v ) a is null, so that: g vv | u =0 = 0 , (4.4) These will agree with k a and (cid:96) a on C whenever C is C . Calling these double-null coordinates is a slight abuse of notation, as we will only require g vv = 0at the horizon u = 0. – 18 –nd we may further fix the gauge: g vi | u =0 = 0 , (4.5)but we cannot require these components to be identically zero in a neighborhood of u = 0, i.e. the derivatives may not vanish. For instance, the inaffinity of k a – given by κ ( v ) = ∂ u g vv | u =0 – cannot be set to zero in general since we have independently fixedthe (cid:96) a vector field, and orthogonality to (cid:96) a defines the constant (or affine) v slices on H + . The extrinsic curvature tensors of constant- v slices are simple in this gauge B ( v ) ij = 12 ∂ v g ij | u =0 , (4.6a) B ( u ) ij = 12 ∂ u g ij | u =0 , (4.6b) χ i = 12 ∂ u g vi | u =0 . (4.6c)where B ( v ) ij and B ( u ) ij denote the null extrinsic curvatures of constant v slices and χ i denotes the their twist. Since H + is achronal, we can specify a new solution to theEinstein equation via a perturbative deformation of the metric on it, so long as the nullconstraint equations are satisfied. In particular, we consider the following perturbationon H + C ( u = 0 , v ≥ ds = − dudv + ( g vv + δg vv ) dv + 2( g vi + δg vi ) dvdy i + ( g ij + δg ij ) dy i dy j . (4.7)The perturbation components δg vv , δg vi , δg ij and their first u derivatives are initialdata that we can freely specify on the characteristic hypersurface u = 0 so long as thisdata satisfies the null constraint equations: δG vv + Λ δg vv = 8 πGδT vv (4.8a) δG uv = 8 πGδT uv (4.8b) δG vi + Λ δg vi = 8 πGδT vi (4.8c)where δG ab denotes the linearized perturbation of the Einstein tensor and Λ is thecosmological constant. Note that the corresponding deformations of the stress energytensor must be sourced by a perturbative modification to the matter fields that itselfsatisfies the fields’ equations of motion on the background geometry. The covariant definition of the inaffinity of k a is k a ∇ a k b = κ ( k ) k b . – 19 –hus we will need to prescribe δg as well as a matter source for it. The latter isaccomplished by a perturbative “stretch”. A nonperturbative stretch is an exponentialrescaling [43, 44]: g (cid:48) ij ( u = 0 , v, y ) = g ij ( u = 0 , ve − s , y ) (4.9a) T (cid:48) vv ( u = 0 , v, y ) = e − s T vv ( u = 0 , ve − s , y ) (4.9b) κ (cid:48) ( v ) ( u = 0 , v, y ) = e − s κ ( v ) ( u = 0 , ve − s , y ) (4.9c)where prime denotes the transformed quantities. Our matter source will be obtained inthe perturbative limit of this, by setting e − s = 1 − (cid:15) , where (cid:15) ∼ O ( δg ) is the parametercontrolling the expansion. Our choice for the full perturbation on v ≥
0, metric andmatter, is then: δg vi ( u = 0 , v, y ) = 0 (4.10a) ∂ u δg vv ( u, v, y ) | u =0 = 2(1 − (cid:15) ) κ ( v ) (0 , v (1 − (cid:15) ) , y ) − κ ( v ) ( u = 0 , v, y ) (4.10b) δg ij ( u = 0 , v, y ) = g ij ( u = 0 , v (1 − (cid:15) ) , y ) − g ij ( u = 0 , v, y ) (4.10c) δT vv ( u = 0 , v, y ) = (1 − (cid:15) ) T vv ( u = 0 , v (1 − (cid:15) ) , y ) − T vv ( u = 0 , v (1 − (cid:15) ) , y ) (4.10d)where in this linearized analysis we will only need to keep track of first order terms in (cid:15) and δg (i.e. we will drop all terms of order (cid:15)δg ). Note that δg vv , ∂ u δg ij , and ∂ u δg vi are allowed to be non-zero. We will see that their values are constrained subject to theabove restrictions.Before we move forward with the analysis, we need to ask whether we can alwaysobtain the stress tensor profile in Eq. (4.10d). This question is difficult to answer inbroad generality. Therefore, from now on we restrict our matter sector to consist of aminimally coupled complex scalar field theory φ (with an arbitrary potential) coupledto some Maxwell field (or consider either separately), with Lagrangian density L matter = − g ac g bd F ab F cd − g ab ¯ ∇ a φ ¯ ∇ b φ ∗ − V ( | φ | ) (4.11)where F ab is the field strength and ¯ ∇ denotes the covariant derivative with respect tothe vector potential A a . Then T vv = 2 ¯ ∇ v φ ¯ ∇ v φ ∗ + F iv F iv , (4.12)Since both φ and A v are free initial data in the characteristic problem, we can sim-ply generate the desired transformation by setting φ (cid:48) ( v, y ) = φ ( ve − s , y ) and A (cid:48) v = e − s A v ( ve − s , y ), and A (cid:48) i = A i ( ve − s , y ).We now proceed to prove that our choice of perturbation solves the null constraintequations with δg vv < H + C . Because the unperturbed spacetime– 20 –atisfies the NEC and no new matter terms are introduced by the perturbation, theperturbed spacetime will likewise satisfy the NEC. For pedagogical clarity, we willfocus on the more illuminating vv -constraint here and relegate the remaining constraintequations to Appendix A. Our analysis here will be twofold: we will first analyze theconstraint (4.8a) in the absence of δg vv , separately compute the contribution of δg vv ,and sum the two together; this is possible so long as we work in the linearized regime.By the Raychaudhuri equation, R vv depends only on the geometry of the u = 0hypersurface: R vv = − ∂ v θ ( v ) − B ( v ) ij B ( v ) ij − κ ( v ) θ ( v ) . (4.13)Therefore in the absence of δg vv (i.e. implementing only the stretch): R (cid:48) vv ( u = 0 , v, y ) = e − s R vv ( u = 0 , ve − s , y ) − (1 − e − s ) θ ( v ) ( u = 0 , v = 0 , y ) δ ( v )= T (cid:48) vv ( u = 0 , v, y ) − (1 − e − s ) θ ( v ) ( u = 0 , v = 0 , y ) δ ( v ) (4.14)where the delta function term results from the discontinuity in θ (cid:48) ( v ) across v = 0. So R (cid:48) vv − T (cid:48) vv = − (1 − e − s ) θ ( v ) [ C ] δ ( v ). We now take the same perturbative limit of thistransformation and re-introduce δg vv : − θ ( u ) ∂ v δg vv − ∇ ⊥ δg vv + χ i ∂ i δg vv + (cid:16) ∇ ⊥ .χ − ∂ v θ ( u ) − B ( v ) ij B ( u ) ij + 8 πG ( − T uv − L matter + F uv ) (cid:17) δg vv − (cid:15)θ ( v ) [ C ] δ ( v ) = 0 (4.15)where all of the quantities multiplying δg vv and its derivatives are background quanti-ties. We offer an alternative derivation of Eq. (4.15) in Appendix B by implementingthe “stretch” using an inaffinity shock [43, 44], which directly induces the delta functionterm in Eq. (4.15).Since by construction, we are only perturbing the data on H + C , δg vv ( u = 0 , v =0 − , y ) = 0, so the delta function term in Eq. (4.15) enforces a jump in δg vv : δg vv ( u = 0 , v = 0 + , y ) = 2 (cid:15) θ ( v ) [ C ] θ ( u ) [ C ] ≤ θ ( v ) [ C ] ≥ θ ( u ) [ C ] < Note that byassumption θ ( v ) [ C ] >
0, and so δg v v <
0, at least in a subset of C . This implies that In a non-generic spacetime, it is possible that θ ( u ) [ C ] = 0, which raises potential concerns about thedivergence in Eq. (4.16). We can avoid this issue by shifting the earliest location of the perturbationto some arbitrarily small v = δ > v = 0. This new cut lies on a past causal horizonthat reaches I and therefore cannot be stationary. In fact, originating the perturbation at v = δ > t i to arbitrarily small values in the source Eq. (2.1) we canaffect the region arbitrarily close to C , but not C itself. With this subtlety in mind, we pick v = 0 asthe origin of the perturbation in the main text because we can get arbitrarily close to C . – 21 –he curve generated by ( ∂ v ) a is nowhere spacelike and at least timelike on a subset of C . In order to open up the lightcone and move the causal surface deeper into the bulk,it would be sufficient if δg vv ≤ everywhere on H + C , not just at C .We will now demonstrate that if δg vv ( u = 0 , v = 0 + , y ) ≤
0, then δg vv ( u = 0 , v, y ) ≤ v >
0, by analyzing the constraint that δg vv satisfies on v > − θ ( u ) ∂ v δg vv − ∇ ⊥ δg vv + χ i ∂ i δg vv + (cid:16) ∇ ⊥ .χ − ∂ v θ ( u ) − B ( v ) ij B ( u ) ij + 8 πG ( − T uv − L matter + F uv ) (cid:17) δg vv = 0(4.17)which we may view as an “evolution” equation for δg vv on u = 0 from which we canderive δg vv on H + C from its value at C .It is not too difficult to see why δg vv ( u = 0 , v, y ) ≤ v > δg vv ( u = 0 , v = 0 + , y ) ≤
0. Suppose δg vv > v . Then, assuming thatall quantities in (4.17) are continuous, there must exist a “last” constant- v slice σ onwhich δg vv ≤ p ∈ σ where δg vv | p = 0(and then immediately becomes positive for larger v ). By construction, we must have ∂ i δg vv | p = 0 and ∇ ⊥ δg vv | p ≤
0. But by (4.17) this implies that ∂ v δg vv | p ≤
0, and so δg vv cannot become positive.This reasoning may seem a bit fast, but it can be made more rigorous (and free ofsimplifying assumptions) using standard techniques. The operator L acting on δg vv in (4.17) is parabolic whenever θ ( u ) <
0; it thus satisfies the weak comparison principlefor parabolic operators, which states that if f and h are functions satisfying L f ≤ L h ≥ f ≤ h on theboundary of the parabolic domain, then f ≤ h everywhere in the parabolic domain.Setting f = δg vv and h ≡
0, we immediately find that L f = 0 and L h = 0, sothe weak comparison principle yields the desired conclusion: δg vv ≤ H + C . Technically, the weak comparison principle is usually stated for domains in R n ,fortunately, it follows as a fairly direct consequence of the maximum principle for ellipticoperators, which does hold for more general manifolds [45]. The functions f and h needonly be of Sobolev type W , ; that is, only their local weak derivatives are requiredto exist [46], which is sufficient for our purposes. In fact, a version of the maximumprinciple for elliptical operators on “rough” null hypersurfaces (including caustics andnon-local intersections specifically on event horizons) was proved in [47].To make sure that our δg vv solution exists, we need to also satisfy the other con-straints (4.8b) and (4.8c). This is easy to do because they are “evolution” equationsfor ∂ u δg ij and ∂ u δg vi on H + C which we can solve no matter what δg vv is. We relegatethis discussion to the appendix. – 22 –et us now discuss possible subtleties in our construction due to caustics. Sincecaustic lines will generically be a measure zero subset of H + [36, 48, 49], we believethat they do not pose a fundamental obstruction to our procedure. Caustic lines canintersect C , at which point C will generically be kinked. At the location of the kink,a chunk of transverse directions, associated to the generators that emanate from thecaustic line, needs to get inserted in the transverse domain on which we place ourboundary data for Eq. (4.17). However, so long as this data satisfies δg vv ≤
0, Eq.(4.17) still guarantee δg vv ≤ H + C . In fact, since these new generatorsdo not extend to the past of the caustics by definition, we expect to have even morefreedom in specifying this boundary data because we do not have to worry about howthis boundary is glued to some past hypersurface.Let us offer an alternative argument to further ameliorate caustic-related worries.As H + C settles down to Kerr-Newman, there exists an earliest cross section µ earliest lying on a past horizon with no caustics in its future. By setting µ earliest as the originof our perturbation (the new v = 0), we can avoid caustics altogether. Furthermore,each perturbation should make the portion of the horizon to the future of µ earliest morestationary, pushing the new µ earliest further to the past eventually approaching H − .Lastly, it is important to show that the perturbation has not shifted the u = 0surface to the point where it is no longer close to the event horizon – especially in theasymptotic region v → ∞ . This is simplest to do if we assume that the background horizon settles down to a stationary spherically symmetric configuration at some finiteaffine parameter, though we expect proper falloffs to hold more generally. The equationfor δg vv then simplifies to: − θ ( u ) ∂ v δg vv − ∇ ⊥ δg vv − ∂ v θ ( u ) δg vv = 0 (4.18)On H + , we have θ ( u ) ∼ v asymptotically. We can then solve for the asymptoticbehavior of δg vv from Eq. (4.18): δg vv ∼ v − (4.19)Therefore u = 0 asymptotes to a stationary null hypersurface after the perturba-tion, so it naturally lines up with the new causal horizon in the v → + ∞ limit. We findthat our proposed perturbation results in a spacetime that solves the Einstein equationand in which the causal horizon is pushed deeper into the bulk unless θ ( k ) [ C ] = 0. We will now use the above perturbation to show that the causal surface C can be movedarbitrarily close to the outermost extremal surface X , the appetizer of the lunch, using– 23 –imple sources only. This requires us to show that (1) the perturbation analyzed inthe previous section can be engineered from simple sources on the boundary, (2) theperturbation can be iteratively repeated both in the past and future, resulting in theapproach of C to X , without incurring high complexity, and (3) that this proceduredoes not change the geometry of the lunch, nor does the causal surface breach the lunchregion in the process. We will begin our discussion by assuming for simplicity thatthe causal surface and the appetizer have the same topology; topological differencesbetween the two surfaces are discussed at the end.To show (1), we simply employ our assumption of the validity of HKLL, discussedin Sec. 1, and evolve the data on H + C ∪ H − C “sideways” towards I to find appropriateboundary conditions at I , which will be smeared local sources. This sideways evo-lution was also used in [24] to prove the simple entropy conjecture in the case wherethe horizon was only perturbatively non-stationary.We will shortly demonstrate (2) in detail. The process is similar in spirit to thezigzag process in Sec. 3, but instead of removing infalling chiral modes, we apply ourperturbation in Sec. 4. First, we will discuss the consequences of repeated iterationsof our perturbation on the future horizon and then add timefolds into our procedure.Let us begin by a comparison between the perturbed and unperturbed causal sur-faces, denoted C (cid:48) and C respectively, as a result of one instance of our perturbation on H + C . Since the perturbation is localized away from the past event horizon H − , it isexpedient to compare the relative location of C with that of C (cid:48) using their position onthe past event horizon: both are slices of H − . Note that in the perturbed geometryno special role is played by C . As shown in Sec. 4, the perturbation guarantees that C (cid:48) is “inwards” (i.e. at larger u ) compared with C so long as θ ( k ) [ C ] (cid:54) = 0. Nothing stopsus from then repeating the perturbation above on the new future horizon. As long assome point on the causal surface satisfies θ ( k ) >
0, the perturbation pushes the causalwedge further inwards.The only obstruction in the construction above occurs if θ ( k ) vanishes identically onthe causal surface. Thus it is clear that the inwards shift of the causal surface obtainedvia simple sources limits to an outermost marginally outer trapped surface µ . (We candefine a rigorous notion of the causal surface approaching arbitrarily close to a surfacewith θ ( k ) = 0 by picking an affine parameterization on H − and defining proximity of No pun intended. We are aware that some mathematician somewhere must be apoplectic with rage after readingthis paragraph. Since we are physicists, we will nevertheless persevere under the usual assumption inholography that HKLL does in fact work out. Concerned readers who do not a priori wish to subscribeto HKLL may take comfort in the following interpretation of our results: we show that reconstructionof the simple wedge is no more complex than that of the causal wedge. – 24 –he two surfaces in terms of the maximal elapsed affine parameter between them.)Let us provide intuition for the existence of µ . Say on H − we can identify twocuts µ and µ such that µ encloses µ and θ ( k ) [ µ ] ≤ θ ( k ) [ µ ] ≥
0. Then we expectan outermost marginally outer trapped surface µ in-between µ and µ . On H − , C plays the role of µ . For µ , we can pick ∂J + [ X ] ∩ H − – whenever it is a full crosssection of H − – which satisfies θ ( k ) ≤ ∂J + [ X ] ∩ H − might be empty if H − falls into a singularity before intersecting ∂J + [ X ].Even so, at least for Kasner-like singularities, we can find cross sections of H − in aneighborhood of the singularity which are trapped [30]. Note also that generically, ourchoices for µ and µ satisfy θ ( k ) [ µ ] < θ ( k ) [ µ ] >
0. We will then have candidates forboth µ and µ , so µ exists.Prima facie the procedure at this point appears to have failed! The causal surfacewill generically stop well away from null separation with X , and even further awayfrom coincidence with X . However, this is only true on the particular timefold underconsideration. To proceed to close the gap further, we reverse the arrow of time. We canthen repeat the procedure above in time reverse to shrink the discrepancy between thecausal wedge and W X . We iterate this procedure via forward and reverse timefolds; eachstep brings the causal surface and X closer. Just like for the JT gravity case in Section3, the causal surface should limit to the outermost extremal surface after sufficientlymany timefolds. Importantly, since the bulk physics involved is entirely classical, thenumber of timefolds required for the causal surface to approach the outermost extremalsurface, within a given precision, should be independent of the Planck scale. This meansthat the complexity of the process cannot diverge in the classical limit N → ∞ . Wetherefore conclude that for X and C of identical topology, the simple wedge and theoutermost extremal wedge coincide.Finally, we address (3): throughout this construction, the geometry of the lunchis left unaltered: the perturbation is localized in the causal complement of the lunch,guaranteeing that the lunch remains undisturbed.What about the case where the topologies are different? This could for examplebe the case in a time-symmetric and spherically symmetric null shell collapse in AdSwhere C is a sphere and X = ∅ [51]. If such topology difference exists, we would expectit to present itself between C and the outermost marginally trapped surface on H − in a timefold (or several timefolds) of our procedure above. Furthermore, because eachiteration of our H + C perturbation pushes the causal surface inwards on H − a bit, atsome point the jump from C to C (cid:48) would have to involve a topology change. As there is This is proved when µ and µ satisfy θ ( k ) [ µ ] < θ ( k ) [ µ ] > D = 4 in [50], but since we can approach H − with such spacelike slices we expect the same to holdfor H − , and the technology used in the extant proof is expected to generalize to higher dimensions. – 25 –othing explicit in our construction that constrains the topology of C (cid:48) according to thatof C , we do not see a fundamental obstruction against such topology changes arisingfrom our iterative procedure. A rigorous treatment of such cases might be interestingbut is left to future work. Having established that the simple wedge is in fact reconstructible using exclusivelysimple experiments, we now explore the implications of our results beyond the converseto the Python’s lunch: what is the dual to the area of the outermost extremal surface?What is the field theory interpretation of our results, and in particular what is the“simple state” dual to the simple wedge?
As noted in the introduction, the simple entropy of [23, 24] is a coarse-graining overhigh complexity measurements conducted after a fixed boundary time t bdy on a singletimefold: S simple [ t bdy , ρ bdy ] = max ρ ∈B S vN [ ρ ] (6.1)where ρ bdy is the actual state of the CFT, t bdy is a choice of boundary time slice, and B is the set of all CFT states (density matrices) that have the same one-point functionsas ρ bdy under any simple sources turned on after the time t bdy (and with some very latetime cutoff to avoid recurrences). That is, B consists of the set of CFT states ρ suchthat (cid:10) E O E † (cid:11) ρ bdy = (cid:10) E O E † (cid:11) ρ , (6.2)for all possible E defined as in Eq. 2.1. The simple entropy at a given boundary time isthus a coarse-graining over high complexity data that preserves all of the simple datato the future (or past) of that time.With these restrictions to a particular subset of boundary time and a fixed timefold,the simple entropy was proposed as a dual to the outer entropy, which is a bulk-definedquantity that coarse-grains over the exterior of an apparent horizon (a surface whichis by definition always marginally trapped ). The outer entropy coarse-grains over allpossible spacetimes that look identical outside of a given apparent horizon to find the A refinement of an apparent horizon that was named a “minimar surface”. Apparent horizons aregenerically minimar surfaces. – 26 –pacetime with the largest HRT surface, and thus the largest von Neumann entropy inthe CFT: S outer [ µ ] = max X Area[ X ]4 G (cid:126) = max ρ ∈H S vN [ ρ ] = Area[ µ ]4 G (cid:126) , (6.3)where X consists of the HRT surfaces of all classical holographic spacetimes containing O W [ µ ], and H is the corresponding set of CFT states; the final equality is provedin [23, 24]. This is done by discarding the spacetime behind µ , constructing a spacetimewith an HRT surface X µ whose area is identical to the area of µ , and then CPTconjugating the spacetime around X µ . By construction, O W [ µ ] is left unaltered.The proposal that the simple and outer entropies are identical says that there isa particular definition of black hole entropy which is a consequence of coarse-grainingover the highly complex physics that we expect describes the interior: S outer [ µ ( t bdy )] = S simple [ t bdy ] (6.4)where t bdy = ∂J − [ µ ] ∩ I . Our construction in Section 4 establishes this conjecture forapparent horizons: in a given timefold, it is possible to push the event horizon all theway up to µ ( t bdy ) without accessing any high complexity data for t > t bdy .Our construction is of course more general, as it applies to timefolds. Extendingthe simple entropy proposal to include timefolds immediately yields the holographicdual to the area of the outermost extremal surface X :Area[ X ]4 G (cid:126) = S outer [ X ] = S simple (6.5)where S simple is obtained from S simple [ t bdy ] by taking t bdy → −∞ and including arbi-trary timefolds. The inclusion of timefolds removes the need for a reference apparenthorizon, and the coarse-grained spacetime (in which the outermost extremal surface isin fact the HRT surface) is obtained by CPT-conjugating around the outermost ex-tremal surface X ; see Fig. 3. Crucially, note that the coarse-graining procedure leavesthe outermost extremal wedge untouched and coarse-grains only over the lunch. Stan-dard entanglement wedge reconstruction via quantum error correction [52–54] appliesto reconstruction of the outermost extremal wedge, since in the coarse-grained space-time obtained by CPT conjugation, the outer wedge of X is exactly the entanglementwedge. Since we will argue below that the coarse-grained spacetime has a simple mod-ular Hamiltonian, entanglement wedge reconstruction using, for example, modular flowas in [55] should be much simpler when based on the coarse-grained state rather thanthe original state. This is consistent with the simplicity of reconstructing the outermostextremal wedge. – 27 – .2 The Simple State So far we have introduced two manipulations that can be done to our original space-time. The first was the zigzag procedure, introduced in Section 5 which makes thecausal wedge coincide with (or become arbitrarily close to) the outermost extremalwedge. We say that the resulting spacetime is ‘exposed’ because everything in the sim-ple wedge can be directly seen by the boundary. The second is the coarse-grainingprocedure introduced above where we CPT conjugate about the outermost extremalsurface and thereby create a state where the outermost extremal wedge coincides withthe entanglement wedge. If we apply both manipulations to the spacetime, we can pro-duce a spacetime where all three wedges (approximately) coincide. This is illustratedin Fig. 6.That is, given any holographic CFT state ρ dual to some entanglement wedge(which will likely have a Python’s lunch), there exists a ρ coarse which is indistinguishablefrom ρ via simple experiments and has no Python’s lunch. Executing our procedurezigzag on this coarse-grained state yields the coarse-grained exposed spacetime, in whichthe causal, simple, and entanglement wedges all coincide or come arbitrarily close tocoinciding. The dual to this is described by the state obtained via the zigzag proceduretogether with the set of simple operators that remains from the final timefold. Weshall refer to this history that includes the the remaining simple operators the simplehistory, and in a slight abuse of notation we will denote this entire history of states as ρ simple .What is the CFT interpretation of ρ simple ? Below we prove that the causal andentanglement wedges coincide exactly if and only if the state dual to the entanglementwedge has a geometric modular flow. The immediate implication is that in the casewhere the zigzag procedure gives an exact coincidence, the simple state has an exactlylocal modular Hamiltonian, and rather than being a history it is in fact a single state.Note that this is suggestive of a CFT dual to a gravitational no-hair theorem. To bemore precise: this result suggests that the set of stationary holographic black holes is tobe identified with the high limited set of states with exactly local modular flow. If coin-cidence between the causal surface and the appetizer is asymptotic rather than exact,then we come to the conclusion that the modular flow generated by the simple state isvery close to local in the sense that only operators with support in the asymptoticallyshrinking region between the causal and entanglement wedges are able to definitivelytell that the two are not identical. Since that region translates (for simple operators) toaccess to arbitrarily late or early times, we find that finite-time simple measurements Note that this definition is unrelated to our definition of an ‘exposed surface’ in the proof ofProposition 1. – 28 – riginal state ρ Coarse-grained state ρ coarse Simple state = coarse-grained exposed state ρ simple Exposed state ρ exposed C X X min XX min C X C CPT X C P T r e f l ec t o n X C P T r e f l ec t o n X Z i g z ag Z i g z ag Figure 6 . A diagram illustrating the relationships between the different states: the originalCFT state ρ , which may be either coarse-grained to obtain ρ coarse by forgetting about highcomplexity operators, or it may be “exposed” by acting on it with simple operators. Thetwo operations modify causally independent and non-intersecting portions of the spacetime,so they commute: after obtaining the coarse-grained state in which X is the HRT surface, wemay perform our zigzag procedure to push the causal surface up to X and obtain the simplestate in which all three wedges coincide. Or, after obtaining the exposed state in which thecausal and outermost extremal wedge coincide, but the entanglement wedge properly containsboth, we may coarse-grain to obtain the same simple state. are unable to tell that the modular flow generated by ρ simple at each stage is not local.The secondary implication is that it is possible to take any holographic state ρ and,via a series of simple operations, render its modular flow (nearly) indistinguishable froma geometric flow via any simple experiments. If the appetizer has sufficient symmetry,then the statement should be true exactly. Let us now prove our theorem, which we– 29 –o in broad generality for boundary subregions. Theorem 1.
Let W C [ R ] denote the causal wedge of a boundary subregion R , and let W E [ R ] denote the entanglement wedge of the boundary subregion. W C [ R ] = W E [ R ] ifand only if the boundary modular Hamiltonian on R generates a geometric flow withrespect to a boundary Killing vector field on R .Proof. Assume that the boundary modular Hamiltonian generates a geometric flowwith respect to some Killing vector field ξ I on ∂M (here I is a boundary spacetimeindex). Under modular flow, a local operator is mapped to another local operator: O ( x, s ) = ρ − is/ πR O ( x ) ρ is/ πR = O ( x ξ ( s )) (6.6)where x ξ ( s ) is the boost along ξ I of x ∈ D [ R ]. Via [55], W E [ R ] can be reconstructedby smearing the modular flow of local operators over D [ R ]. Since in this case modularflow is an automorphism on the space of local operators in D [ R ], we can reconstructall operators in W E [ R ] by smearing local operators:Φ( x ) = (cid:90) D [ R ] f ( X | x ) O ( X ) dx, (6.7)where f ( X | x ) is a smearing function. If there is a gap between W E [ R ] and W C [ R ],then operators that are localized to the gap should commute with all local operatorson the boundary (within our code subspace) via the extrapolate dictionary (note thatthis only works in the large- N limit where we don’t have to worry about gravitationaldressing). However this is inconsistent with the equation above; so x ∈ W C [ R ]. Butthis argument holds for all local operators: there exist no local operators in the gapbetween W X [ R ] and W C [ R ]. In the large- N limit (without backreaction), this meansthat there simply is no gap between the two wedges: W X [ R ] = W C [ R ].To prove the other direction, we consider the proof of [55] for the zero-mode formulaof entanglement wedge reconstruction (appendix B.1 of [55]). Starting with equationB.71, it is shown that the nonlocality of modular flow on ∂W E [ R ] is due to the change inthe instrinsic metric of spatial slices of ∂W E [ R ] (in particular, the loss of ultralocality).When ∂W E [ R ] is stationary, the metric on codimension-two slices does not change withevolution along the congruence. This means that the modular flow on ∂W E [ R ] is local,which in turn implies that the boundary modular flow is local as well. Our primary technical result in this article is the proof of the converse to the Python’slunch proposal in the strict large- N limit: operators that lie outside of a Python’s– 30 –unch are simply reconstructible in the dual CFT, and moreover this reconstructiononly relies on the bulk dynamics in the large- N limit, manifestly respecting the causalstructure of the background metric.We emphasize that bulk reconstructions that are causal in this sense cannot workfor the interior of the Python’s lunch because no causal horizon can intersect the lunch.The CFT encoding of the Python’s lunch appears to involve highly non-local quantumgravity effects. An example of such non-local dynamics is the ER=EPR conjecture [56,57] which asserts that the entanglement between an evaporating black hole and itsHawking radiation after the Page time must allow complicated operations on the distantradiation to change the state behind the lunch, drastically violating the naive causalstructure dictated by the background metric. It has been speculated that wormhole-like “corrections” to the background geometry connecting the radiation to the blackhole interior could explain the “true” causal structure not captured by the backgroundmetric. It is natural to speculate that similar dynamics are at play in the Python’slunch encoding into the boundary.We will now discuss various generalizations of our main results: We have focused here primarily on compact surfaces, but we may pose similar questionsfor boundary subregions: given the state ρ R on a boundary subregion R , how complexis the reconstruction of operators behind the event horizon but within the outermostextremal wedge? This requires a treatment of surfaces with a boundary-anchored com-ponent rather than surfaces whose components are all compact. Most of our resultsgeneralize almost immediately to the boundary-anchored case: Lemma 3 [37] makesno reference the topology of surfaces (beyond the homology constraint); similarly forthe proofs of Lemmas 1 and 2. The perturbed initial data prescription at the causalsurface also carries over mutatis mutandis. As noted in Section 4, the weak comparisonprinciple operates on the basis of the maximum principle for elliptic operators. Thelatter does indeed apply to bounded domains in general, and to boundary-anchoredhypersurfaces in AdS particular (see [58] for a discussion in the context of AdS/CFT).The main potential source of difficulty is the falloff: both the causal surface and theoutermost extremal surface approach the asymptotic boundary, but not with the sametangent space; the asymptotic falloff of δg must approach zero sufficiently fast so as tonot spoil the asymptotics while bridging the gap between the two surfaces. We expectthat the appropriate falloff conditions can be satisfied, but this remains a subject forfuture work. – 31 – .2 Robustness under Quantum Corrections Fundamentally, the classical calculations done in this paper are only interesting asan approximation to the fully quantum dynamics that actually describe the bulk inAdS/CFT. Do our arguments extend to the semiclassical setting where the backgroundspacetime is still treated classically, but with quantum fields propagating on it? Dothey generalize to the regime where perturbative corrections to the geometry, sup-pressed by powers G (cid:126) , are allowed to contribute? A number of important assumptionsbreak down in this case: Raychaudhuri’s equation is still valid, but the null energycondition will not generally hold, and so light rays emanating from a classical extremalsurface can defocus. Fortunately, the quantum focusing conjecture (QFC) states thatthe generalized entropy of null congruences emanating from QESs is always subject tofocusing. In a semiclassical or perturbatively quantum bulk, the appetizer is the out-ermost quantum, not classical, extremal surface; the QFC ensures that the outermostquantum extremal wedge always contains the causal wedge. The question is whetherwe can still expand the causal wedge using appropriate sources and timefolds in orderto bridge the gap between the causal surface and the appetizer.This is a much harder question than the classical question discussed here: the classof allowed QFT states is simply much harder to classify and use than classical fieldtheory states. However, in particularly simple examples, for instance where the causalwedge is approximately Rindler-like, a quantum version of the “left stretch” appearsto be well defined and gives exactly the right change in energy to reduce focusingand remove the perturbatively small, i.e. O (1 /N ), distance between the causal andoutermost extremal wedges [25]. It is feasible that in the limit of many zigzags, thecausal wedge would approach a Rindler-like region, at which point it becomes possibleto apply the bulk unitaries discussed in [25] to eliminate the remaining small gap. Weleave a detailed study of this to future work. In asymptotically flat spacetimes, asymptotic infinity is lightlike rather than timelike.However, there do not seem to be any major obstructions to adapting the results ofthis paper to that setting. Instead of the timefolds of the asymptotically AdS problem,in asymptotically flat space one would presumably evolve forwards along future nullinfinity, then backwards with different boundary conditions that remove focusing atthe past event horizon, in order to produce a state where the causal surface is very Apparent counterexamples are only possible when the time evolution includes interaction withan auxiliary system; once you include these auxiliary interacting modes when defining quantum ex-tremality then the apparent causality issues go away. – 32 –lose to a past apparent horizon. Then one would evolve backwards and forwards alongpast null infinity in order to produce a state where the causal surface is very closeto a future apparent horizon. At each step, the causal wedge increases in size. Aftersufficiently many such timefolds, the causal surface should approach the outermostextremal surface, as in the asymptotically AdS case. The interpretation of our resultsin the asymptotically flat case is naturally obfuscated by relatively inchoate status of flatholography. We may speculate that extremal surfaces are important more generally fordefining entropy in gravity; it is also possible that a similar notion of a Python’s lunchapplies beyond AdS holography. We do not subscribe to any particular interpretation– here we simply note that the technical aspects of this work are likely not restrictedto AdS.
Let us finish with a few comments on cosmic censorship, a prima facie unrelated conjec-ture about classical General Relativity. (Weak) cosmic censorship [38] is essentially thestatement that high curvature physics lies behind event horizons. One of its landmarkconsequences is that trapped surfaces lie behind event horizons, and that consequentlymarginally trapped and in particular extremal surfaces lie on or behind event hori-zons [36]. It is clear from the above discussion that any violation of cosmic censorshipwould be quite problematic for the Python’s lunch picture: if the nonminimal extremalsurface could lie outside of the event horizon (or, in the quantum case, could com-municate with I ), then operators behind would lie properly within the causal wedgeand would thus be reconstructible by HKLL despite being exponentially complex. ThePython’s lunch proposal thus appears to depend heavily on the validity of cosmic cen-sorship – which is known to be violated in AdS [59–61]. As matters currently stand,violations of cosmic censorship notwithstanding, it is possible to prove that the holo-graphic entanglement entropy prescription guarantees that trapped surfaces must liebehind event horizons [41]. We could however have proven the same statement fromholographic complexity: marginally trapped surfaces (and therefore, trapped surfacesalso, by the reasoning of [41]) must lie behind event horizons, for if they did not, op-erators behind the Python’s lunch could be reconstructed in a simple procedure. Thissuggests that in AdS/CFT, aspects of cosmic censorship may be reformulated as “com-plexity censorship”: that high complexity physics must be causally hidden and thusunable to causally communicate to I . – 33 – cknowledgments It is a pleasure to thank S. Alexakis, R. Bousso, S. Fischetti, L. Susskind for helpfuldiscussions. NE is supported by NSF grant no. PHY-2011905, by the U.S. Departmentof Energy under grant no. DE-SC0012567 (High Energy Theory research), and byfunds from the MIT physics department. GP is supported by the UC Berkeley physicsdepartment, the Simons Foundation through the ”It from Qubit” program, and alsoacknowledges support from a J. Robert Oppenheimer Visiting Professorship. ASM issupported by the National Science Foundation under Award Number 2014215.
A uv and vi constraints
Here we write down the general structure of the perturbative equations δG uv = 8 πGδT uv and δG vi = 8 πGδT vi :12 g ij ∂ v ∂ u δg ij = ( ˆ L ) ij ∂ u δg ij + ( ˆ L ) i ∂ u δg vi + ˆ L δg vv + 8 πGδT uv (A.1a) ∂ v ∂ u δg vi = ( ˆ L (cid:48) ) j ∂ u δg ij + ˆ L (cid:48) ∂ u δg vi + ( ˆ L (cid:48) ) i δg vv + 8 πGδT vi (A.1b)where all ˆ L n and ˆ L (cid:48) n represent linear differential operators that depend only on thebackground metric and do not have u derivatives. The only significance of Eqs. (A.1a)and (A.1b) for us is that they are consistent with the δg vv solution to Eq. (4.17).Plugging in that solution reduces Eqs. (A.1a) and (A.1b) to coupled linear differentialequations involving ∂ u δg ij | u =0 and ∂ u δg vi | u =0 which are first order in ∂ v . Note that theinitial conditions at v = 0 are ∂ u g ij ( u = 0 , v = 0 − , y ) = ∂ u g vi ( u = 0 , v = 0 − , y ) = 0, asenforced by the perturbation being localized away from H − . B Alternative derivation of the vv constraint
Here we provide an alternative derivation for the same perturbation discussed in Sec.4. Instead of the transformations (4.9a), (4.9b), and (4.9c) on H + C , we can equivalentlyinsert an inaffinity shock at v = 0 [43, 44]: κ ( v ) = (1 − e − s ) δ ( v ) (B.1)and take the (1 − e − s ) ∼ (cid:15) limit. In addition, we want to introduce the following δg ab transformation: ds = − dudv + ( g vv + δg vv ) dv + 2( g vi + δg vi ) dvdy i + ( g ij + δg ij ) dy i dy j , (B.2)– 34 –ith δg vi | u =0 = 0 (B.3) δg ij | u =0 = 0 (B.4) ∂ u δg vv | u =0 = 0 (B.5)We need to apply the combined perturbations in Eqs. (B.1) and (B.2) to the vv constraint: δG vv + Λ δg vv = 8 πGδT vv (B.6)The only contribution in δG vv from Eq. (B.1) is through the κ ( v ) θ ( v ) term in Eq. (4.13).Summing this up with the contribution from the contribution from δg ab of Eq. (B.2),we get: − θ ( u ) ∂ v δg vv − ∇ ⊥ δg vv + χ i ∂ i δg vv + (cid:16) ∇ ⊥ .χ − ∂ v θ ( u ) − B ( v ) ij B ( u ) ij + 8 πG ( − T uv − L matter + F uv ) (cid:17) δg vv − (cid:15)θ ( v ) [ C ] δ ( v ) = 0 (B.7)Similarly, the uv and vi constraints could be analyzed resulting in Eqs. (A.1a) and(A.1b). References [1] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield,
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