PHYSICAL REVIEW D 102, 086021 (2020)
Ensemble from coarse graining: Reconstructing the interior of an evaporating black hole
Kevin Langhoff1,2 and Yasunori Nomura 1,2,3 1Berkeley Center for Theoretical Physics, Department of Physics, University of California, Berkeley, California 94720, USA 2Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
(Received 20 August 2020; accepted 23 September 2020; published 27 October 2020)
In understanding the quantum physics of a black hole, nonperturbative aspects of gravity play important roles. In particular, huge gauge redundancies of a gravitational theory at the nonperturbative level, which are much larger than the standard diffeomorphism and relate even spaces with different topologies, allow us to take different descriptions of a system. While the physical conclusions are the same in any description, the same physics may manifest itself in vastly different forms in descriptions based on different gauge choices. In this paper, we explore the relation between two such descriptions, which we refer to as the global gauge and unitary gauge constructions. The former is based on the global spacetime of general relativity, in which understanding unitarity requires the inclusion of subtle nonperturbative effects of gravity. The latter is based on a distant view of the black hole, in which unitarity is manifest but the existence of interior spacetime is obscured. These two descriptions are complementary. In this paper, we initiate the study of learning aspects of one construction through the analysis of the other. We find that the existence of near empty interior spacetime manifest in the global gauge construction is related to the maximally chaotic, fast scrambling, and universal dynamics of the horizon in the unitary gauge construction. We use the complementarity of the gauge choices to understand the ensemble nature of the gravitational path integral in global spacetime in terms of coarse graining and thermality in a single unitary theory that does not involve any ensemble nature at the fundamental level. We also discuss how the interior degrees of freedom are related with those in the exterior in the two constructions. This relation emerges most naturally as entanglement wedge reconstruction and the effective theory of the interior in the respective constructions.
DOI: 10.1103/PhysRevD.102.086021
I. INTRODUCTION AND GRAND PICTURE This string scale dynamics seems to exhibit a variety of universal behaviors as explored in Refs. [4–8]. A black hole is a special object. When described in the These two seemingly different pictures, however, can global spacetime of general relativity, it has a spacetime be different manifestations of the same physics [9–11]. region from which nothing can escape to spatial infinity The basic idea is that the two pictures give equivalent without encountering a singularity. When viewed from a physical conclusions after taking into account huge gauge distance, the large redshift caused by a strong gravitational redundancies of gravitational theory at the nonperturba- field makes the density of states exponentially large even at tive level [12,13], which are much larger than the standard the quantum level [1,2], while at the same time a classically diverging redshift factor makes the intrinsic scale of diffeomorphism and relate even spaces with different dynamics reach the string scale near the horizon, causing topologies [12,14]. The special features of the ultraviolet the breakdown of the classical spacetime description [3]. (UV) string dynamics manifest in one way of fixing the redundancies are related with the existence of large interior spacetime in the infrared (IR) manifest in another fixing [9,10]. The two constructions are complementary. In the construction based on global spacetime, one can use Published by the American Physical Society under the terms of semiclassical techniques to understand certain important the Creative Commons Attribution 4.0 International license. – Further distribution of this work must maintain attribution to aspects of the black hole physics [15 19]. In the other the author(s) and the published article’s title, journal citation, construction based on a distant view, one may find a and DOI. Funded by SCOAP3. restriction on the applicability of semiclassical theory
2470-0010=2020=102(8)=086021(22) 086021-1 Published by the American Physical Society KEVIN LANGHOFF and YASUNORI NOMURA PHYS. REV. D 102, 086021 (2020) based on the (non)existence of approximate linear oper- actually not independent, and as a result the von Neumann ators representing the excitations [11]. entropy of Hawking radiation follows [15–17] the Page The gist of this paper is to explore the relation between curve [30]. It is plausible that a part of this reduction of state the two constructions described above and initiate the study space is associated with the nonperturbative gravitational of learning aspects of one construction through the analysis gauge redundancies [12–14], although a part of them may of the other. From the quantum gravity point of view, a not. This reduction also implies that, for an old black hole, black hole is special in that an appropriate treatment of the the degrees of freedom in the interior are not independent of nonperturbative gauge redundancies is vital in obtaining the those in the exterior, as anticipated earlier [31]. correct physics (although a similar situation may also occur A puzzling feature of this analysis is that the path integral in cosmology [20,21]). It is, therefore, particularly impor- gives tant in the black hole physics not to conflate pictures deriving from different gauge fixings. We thus begin our hψ Ijψ Ji¼δIJ; ð2Þ discussion with short descriptions of the two constructions based on two different ways of fixings the nonperturbative which is inconsistent with Eq. (1) if both are taken at face gauge redundancies. value. Reference [18] interpreted this to mean that the gravitational path integral actually computes the average of A. “Global gauge” construction a quantity over some ensemble, which makes the two equations compatible. This, however, brings the question: This construction starts from the conventional global what ensemble does the path integral represent? In this spacetime picture of general relativity. Perturbative quan- paper, we will discuss how the relevant ensemble can tization may begin with equal-time hypersurfaces that emerge in a consistent unitary theory of gravity. extend smoothly to both the exterior and interior of the Summarizing, the basic idea of the construction is to black hole [22,23]. In this picture, the black hole manifestly start from a description that is highly redundant but is has the interior region, as implied by general relativity. based on familiar global spacetime of general relativity. The problem, however, is to see its compatibility with The price to pay is that in order to see truly independent unitarity [24–26]. First, explicit semiclassical calculation degrees of freedom—and hence unitarity of the theory— seems to indicate that radiation emitted from the black hole one must take into account huge nonperturbative gravi- is maximally entangled with modes in the interior, violating tational gauge redundancies, including those relating the unitarity of the S matrix [24]. Second, even if we different spatial topologies [12–14].Ifthisisdonecare- postulate that Hawking radiation somehow carries infor- fully, however, one can see the correct physics as mation about collapsing or infalling matter to save unitarity, demonstrated recently [15–19]. it then leads to the problem of cloning: the information about fallen matter is duplicated to Hawking radiation, “ ” violating the no-cloning theorem of quantum mechanics B. Unitary gauge construction [27]. Finally, depending on hypersurfaces one chooses, the An alternative construction is to start from a manifestly interior of the black hole can be viewed as having an ever- unitary description [3,30,32]. This naturally arises in a increasing spatial volume [28,29]. The existence of such “distant description” of a black hole. When viewed from a large space does not seem to be consistent with the distance, the black hole carries Hawking cloud around it, Bekenstein-Hawking entropy [1,2], bounding the number whose intrinsic energy scale (local, or Tolman [33,34], of independent black hole states by the horizon area. temperature) increases toward the horizon because of large Recently, there has been significant progress in address- gravitational blueshift. This scale reaches the string scale ing these issues. In particular, new saddles in the gravita- on a surface a microscopic distance away from the classical tional path integral, called replica wormholes, were horizon. This surface is called the stretched horizon [3],at discovered which contribute to the calculation of entropies which the semiclassical description of spacetime breaks [18,19]. Because of this contribution, naively orthogonal down. Since the dynamics around the stretched horizon black hole microstates jψ Ii (I ¼ 1; …; K) develop overlaps cannot be described by a low-energy theory, one may of the form consistently assume that the physics there, which is responsible for the Hawking phenomenon, is unitary. 2 −S0 jhψ Ijψ Jij ¼ δIJ þ Oðe Þ; ð1Þ A virtue of this picture is that it is intuitive. The evolution is unitary by construction (or one can say, as suggested by where S0 is the coarse-grained entropy of the system, in this the AdS=CFT correspondence [35–37]), and the interpre- case the black hole. While these overlaps are exponentially tation of the Bekenstein-Hawking entropy is simple: it suppressed, for K >eS0 they add up and after appropriate basically comes from qubits of a Planck density on the “diagonalization” lead only to eS0 independent states; all stretched horizon. Most of the gravitational gauge redun- the other states are either null or not independent. Due to dancies are regarded as being fixed, leaving only the this phenomenon, seemingly independent interior states are standard diffeomorphism and perhaps a little more.
086021-2 ENSEMBLE FROM COARSE GRAINING: RECONSTRUCTING THE … PHYS. REV. D 102, 086021 (2020)
The issue in this picture is to understand how the microstates. In this paper, we will see that this also interior spacetime of a black hole, whose existence is elucidates the ensemble nature of the gravitational path implied by the equivalence principle, may emerge integral in the global gauge construction. [26,38,39]. In this description, the stretched horizon The effective theory erected over the state of the system behaves—in a sense—as a surface of regular material at a given time (without invoking boundary time evolution such as a piece of coal. While this makes unitarity in the language of holography [47–49]) describes only the manifest, the existence of interior spacetime is obscured; causal region associated with the black hole zone at that we know that the surface of a piece of coal does not allow time [9], giving a specific realization of the idea of black for an object to fall freely into it. What is special about the hole complementarity [3,50]. For an old black hole, the stretched horizon, making it distinct from a regular erection of the effective theory must involve Hawking material surface? radiation emitted earlier, as suggested in Ref. [31], although This problem was addressed in Refs. [9–11].Thereare the detailed realization is different from that contemplated two important characteristics that emerge when a black there: the degrees of freedom describing the second exterior hole is formed. First, because of large gravitational red- of the effective two-sided geometry must also involve black shift, energy gaps between different black hole micro- hole “soft mode” degrees of freedom [9–11,51]. The states become exponentially small, as measured in the nonlocal identification of degrees of freedom needed here asymptotic region. This means that the density of states is understood to come from gauge fixing adopted by the becomes exponentially large [1,2]. Second, the intrinsic unitary gauge construction. energy scale (the local temperature of Hawking cloud) To analyze the relation between this picture and the becomes large, of order the string scale, near the stretched results of Refs. [15–19] adopting entanglement wedge horizon. The dynamics at such energies is believed to be reconstruction [52–57], we need to take into account the maximally quantum chaotic [4], fast scrambling [5,6],and effect of time evolution [11]. As we will see in this paper, not to have a feature discriminating low-energy species, this reveals a structure of bulk reconstruction that is not such as a global symmetry [7,8]. These dynamical features manifest in the simplest entanglement wedge considera- play a crucial role in the emergence of the interior tion: the amount of information one can reconstruct from spacetime. radiation at a given time depends on the spacetime location Because of the dynamical properties described above, within the entanglement wedge. This feature originates the state of a black hole quickly becomes a typical state in from the fact that the reconstruction employs a quantum the relevant microcanonical ensemble, which treats all information theoretic protocol [5] which, unlike the erec- low-energy species in a universal manner. The prescrip- tion of an effective interior theory, uses boundary time tion in Refs. [9–11] then allows us to erect an effective evolution. We elucidate this relation in detail and discuss theory of the interior on each microstate, which can properties of the entanglement wedge reconstruction that describe the fate of an object falling into the black hole. are contrasted with those of the effective interior theory, This construction does not work for a regular material which uses the black hole soft modes in addition to surface because of the lack of the typicality and univer- radiation degrees of freedom. sality, hence singling out the stretched horizon. An important point is that the construction of the effective C. Outline of the paper theory is restricted to the modes in the near black hole In this paper, we mainly analyze the unitary gauge ω region, called the zone, whose characteristic frequencies construction to understand features of the global gauge 10 are sufficiently, e.g., of Oð Þ,largerthantheHawking construction. In Sec. II, we begin with discussion of the temperature TH. This structure makes it possible, unlike unitary gauge construction, reviewing relevant aspects the scenarios discussed in Refs. [40–46], that the oper- of the description in Refs. [9–11].InSec.III,wefind ators describing the interior are state independent, i.e., that this construction exhibits the same ensemble standard linear operators defined throughout the space of property as that found in Ref. [18], providing an microstates, up to exponentially suppressed corrections of understanding of an origin of the ensemble nature of −ω order e =TH.Infact,Ref.[11] argued that the existence of the gravitational path integral. In Sec. IV, we investigate such globally defined operators is necessary for the reconstruction of the interior. We discuss two possible emergence of semiclassical physics, which has an intrinsic ways to reconstruct the interior—effective theories of −ω ambiguity of Oðe =TH Þ. the interior and entanglement wedge reconstruction— An erection of an effective theory of the interior and study their relation. Finally, we conclude in Sec. V. involves coarse graining: many different states in the The two Appendixes contain details of the stretched microscopic theory correspond to a single state in the horizon, soft modes, and the analysis of signal propa- effective theory. This coarse graining is the root of gation and scrambling times. the apparent uniqueness of the infalling vacuum, despite Throughout the paper, we focus on a spherically sym- the existence of exponentially many black hole metric, non-near extremal black hole in asymptotically flat
086021-3 KEVIN LANGHOFF and YASUNORI NOMURA PHYS. REV. D 102, 086021 (2020) or AdS spacetime.1 We adopt natural units c ¼ ℏ ¼ 1, and A. Semiclassical spacetime from approximate we use ls to denote the string length. state independence
Consider the space HM of pure states in which the II. UNITARY GAUGE CONSTRUCTION energy E in a spatial region is bounded by E 1 unitary over the relevant timescale. This operator does not We, however, expect that our basic argument applies to a near commute with the boundary Hamiltonian and induces extremal black hole as well. We also expect that a similar analysis applies to the cosmological horizon of de Sitter spacetime, where components whose energies measured in the asymptotic the global gauge and unitary gauge constructions correspond to region are lower than that of the corresponding vacuum the treatments in Refs. [58,59] and [10,20], respectively. state. However, the evolution still brings the original 2 In the distant description, the distribution of the degrees of vacuum states to the vacuum states in the interior, and freedom associated with the black hole (soft modes) is peaked toward the stretched horizon, so one may say “on and outside the so for the excited states as well, preserving the atypicality stretched horizon” instead of “outside the stretched horizon.” In of the excited states at the microscopic level. this paper, we adopt the latter for brevity. This atypicality of excited states is in stark contrast with 3The boundary here need not be the conformal boundary of the claim in Ref. [46], which is derived from the picture in AdS or an asymptotic infinity in flat spacetime but can be – – Refs. [40 42]. In particular, unlike the scenarios considered the holographic screen in Refs. [60 62], although in the latter – case a fully unitary description may require an ad hoc intro- in Refs. [40 46], the structure described above implies that H duction of extra ingredients, such as a superposition of different the Hilbert space for semiclassical excitations, exc, built geometries. on each of the orthogonal microstates of the black hole and 086021-4 ENSEMBLE FROM COARSE GRAINING: RECONSTRUCTING THE … PHYS. REV. D 102, 086021 (2020) FIG. 1. The Fock spaces built on microstates of a black hole of mass M, which dominate the microcanonical ensemble HM, do not significantly overlap with each other because of the energy restriction imposed on hard modes. Furthermore, the collection of excited H H states, BδE M, forms only an exponentially small subset of the microcanonical ensemble, MþδE, of the same energy. radiation emitted from it need not overlap significantly with We find that the exponential factor associated with the each other. Indeed, from genericity consideration, we microstate entropies indeed disappears. However, the −δ expect that states representing the same semiclassical exponential factor e E=TH associated with the energy of excitation but built on different orthogonal microstates A the excitation remains. This implies that Fock spaces of and B have overlap hard modes built on different orthogonal microstates are orthogonal up to corrections exponentially suppressed in ðAÞ† ðBÞ 1 − δE δ Ψ O O Ψ ≈ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2T E=TH. This allows us to treat the microscopic Hilbert Ah ðMÞj δE δE j ðMÞiB O p e H ; eSbhðMÞþSrad space as ð6Þ H ≈ H ⊗ H exc vac; ð8Þ where jΨðMÞi represent the microstates of the black A;B H hole and radiation, on which semiclassical theories are where the elements of vac cannot be discriminated as – quantum degrees of freedom in semiclassical theory. In built [9 11], and SbhðMÞ and Srad are the coarse-grained entropies of the black hole and radiation, respectively. other words, semiclassical theory is the theory describing the physics associated with the H factor, which is OðA;BÞ exc δE are the suitably normalized operators that excite an insensitive to the microscopic physics occurring in the appropriately smoothed semiclassical mode of energy δE in H vac part. The structure of Hilbert space described here is the black hole region. depicted in Fig. 1. While the overlap in Eq. (6) is suppressed by the large − 2 The structure of the Hilbert space given above makes it exponential factor e ðSbhðMÞþSradÞ= , this by itself is not possible that for an operator OIJ in the semiclassical theory, enough to suppress the overlap between Fock spaces built ˆ we can define global operator OIJ in the microscopic on different microstates. To see this, we can consider the Hilbert space which acts linearly throughout the space of all OðAÞ Ψ probability for the state δE j ðMÞiA to overlap with the microstates: corresponding semiclassical state built on some other 4 Ψ S ðMÞþS microstate orthogonal to j ðMÞiA : e bhX rad Oˆ O Ψ Ψ IJ ¼ IJj IðMÞiAAh JðMÞj; ð9Þ eSbhXðMÞþSrad A¼1 ðAÞ† ðBÞ 2 − δE Ψ O O Ψ ≈ T jAh ðMÞj δE δE j ðMÞiBj Oðe H Þ: ð7Þ B¼1;B≠A where A runs over orthogonal vacuum microstates, I and J are the indices specifying semiclassical states (regardless of Ψ 4 Ψ the microstate), and j IðMÞiA is the semiclassical state I Recall that j ðMÞiA;B represent microstates of the black hole and radiation with the black hole put in the semiclassical vacuum, built on microstate A. In Ref. [11], it was conjectured that so that a generic state in the Hilbert space of dimension the emergence of semiclassical physics requires the exist- 5 eSbhðMÞþSrad has the black hole of mass M. Note that since black ence of these approximately state-independent operators. hole evaporation is a thermodynamically irreversible process These operators obey the algebra of the semiclassical [65,66], most of these microstates do not become a state with a larger black hole in empty space when evolved backward in time—there is always junk radiation around it. This, however, 5This can be viewed as an extension of an analogous leading- does not change the fact that there are eSbhðMÞþSrad independent order statement about the geometry [67,68] to higher order in the microstates relevant for the discussion here. Newton constant expansion, including matter excitations. 086021-5 KEVIN LANGHOFF and YASUNORI NOMURA PHYS. REV. D 102, 086021 (2020) theory throughout the microstates, up to corrections of the species, frequency, and angular-momentum −Eexc=TH order e , where Eexc is the energy of the semiclassical quantum numbers of a mode, and En is the energy excitation. These corrections are the intrinsic ambiguity of of the state jfnαgi as measured in the asymptotic Δ the semiclassical theory, requiring Eexc to be sufficiently region (with precision ). ðnÞ larger than TH. (ii) jψ i are orthonormal states of the soft modes in entangled with jfnαgi. Because of the energy con- B. Black hole interior from chaotic dynamics straint imposed on the black hole, these states have at the horizon energies M − En, with precision Δ.Assumingthatthe Another important ingredient of the description in density of hard mode states is negligible compared Refs. [9–11] is the proposed UV-IR relation, in particular with that of the soft modes (which is justified as we are only interested in states that do not yield significant the role chaotic dynamics at the horizon plays for the 7 emergence of interior spacetime. backreaction), the density of soft mode states is given It is widely believed that the dynamics of the stretched by the standard Bekenstein-Hawking formula, imply- horizon in a distant picture is maximally quantum chaotic ing that in runs over the n-dependent range [4]. It is also expected that this dynamics does not respect S ðM−EnÞ any global symmetry [7,8]. The claim of Refs. [9–11] is in ¼ 1; …;e bh : ð13Þ that these properties of the stretched horizon dynamics are crucial for the emergence of interior spacetime. In fact, With the black hole put in the semiclassical vacuum, together with the exponential degeneracy of states, the any extra attribute a hard mode state may have, e.g., a appearance of such dynamics can be viewed as the quantum charge or angular momentum, is compensated by that mechanical analog of gravitational collapse in classical of the corresponding soft mode states (within the general relativity. precision allowed by the uncertainty principle). This To see how this works, consider a system with a black implies that soft mode states associated with different hole, whose mass M is determined by the maximal hard mode states are virtually orthogonal: precision of hψ ðmÞjψ ðnÞi¼δ δ : ð14Þ Δ ≈ 2π im jn mn imjn Oð THÞð10Þ ϕ allowed by the uncertainty principle. At a given time t, the (iii) j ai represents the set of orthonormal states repre- state of the system—with the black hole being put in the senting the system in the far region r>rz.We semiclassical vacuum—is given by assume that these are fully specified by the states of Hawking radiation emitted earlier, i.e., emitted from S ðM−EnÞ S ≈ X e bhX Xe rad r rz to the asymptotic region before time t, ðnÞ jΨðMÞi ¼ c jfnαgijψ ijϕ i; ð11Þ although this assumption is not essential. S in nina in a rad n in¼1 a¼1 Eq. (11) is then the coarse-grained entropy of the early radiation. where the state is assumed to be normalized: The spatial distribution of the soft modes is determined by the blueshifted local Hawking temperature; see Appendix A. − X eSbhXðM EnÞ XeSrad It is peaked toward the stretched horizon, at which the local c 2 1: 12 j ninaj ¼ ð Þ temperature is of order the string scale and the dynamics is 1 1 8 n in¼ a¼ maximally chaotic. We thus expect that the coefficients c in Eq. (11) take generic values in the spaces of the hard The three factors in the right-hand side of Eq. (11) represent nina and soft modes. In particular, statistically on average states of the hard modes, soft modes, and modes outside the zone region (far modes), respectively. The notations and 1 basic properties for these states are the following6: ∼ ffiffiffiffiffiffiffi jcninaj p ; ð15Þ (i) jfnαgi are orthonormal states of the hard modes, Stot with n ≡ fnαg representing the set of all occupation numbers nα (≥0). The index α collectively denotes 7 In fact, we expect that the logarithm of the dimension of the hard mode Hilbert space, ln dim Hhard, is much smaller than 6 − By construction, the system of a black hole and radiation SbhðM EnÞ and Srad throughout the history of the black hole. evolves unitarily with the state taking the form of Eq. (11) at 8Recall that this dynamics cannot be described by a low- each moment in time. In particular, the entanglement entropy energy theory. In fact, the internal dynamics near the stretched vN vN between the black hole and radiation Shardþsoft ¼ Srad follows the horizon is expected to be nonlocal in the spatial directions along vN Page curve [30], where SA is the von Neumann entropy of the horizon [5,6]. This nonlocality arises presumably from gauge subsystem A. fixing performed to go to the unitary gauge description. 086021-6 ENSEMBLE FROM COARSE GRAINING: RECONSTRUCTING THE … PHYS. REV. D 102, 086021 (2020) where statistical fluctuations over different microstates. The nor- malized coarse-grained state kfnαg⟫ is thus given by X X −En SbhðM−EnÞ Srad T SbhðMÞ Srad S ≡ e e ¼ e H e e . sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − tot X eSbhXðM EnÞ XeSrad En Em n n 2 − ðnÞ kfnαg⟫ ¼ e TH e TH c jψ ijϕ i; ð20Þ nina in a ð16Þ m in¼1 a¼1 − − 2 c ðSbhðM EnÞþSradÞ= With nina being complex numbers, their phases are uni- up to a fractional correction of order e ∼ − 2 c Stot= formly distributed and the variance of nina is comparable to e in the overall normalization. the average of jcninaj. Furthermore, because of the fast With this coarse graining, the state of the system in scrambling nature of the dynamics [5,6], these configura- Eq. (11) can be written as tions are reached quickly. The standard thermal nature of X 1 En the black hole is then obtained upon tracing out the soft −2 kΨðMÞ⟫ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e TH jfnαgikfnαg⟫ ð21Þ modes: P −Em T n me H Ψ Ψ Trsoftj ðMÞih ðMÞj X regardless of the values of cnina, which takes the form of the 1 −En e TH nα nα ⊗ ρϕ ; 17 standard thermofield double state of the two-sided black ¼ P −Em jf gihf gj ;n ð Þ T me H n hole [69,70]. We can therefore build the effective theory describing the interior on this state, as described in where ρϕ;n are reduced density matrices for the early Refs. [9–11] and will be discussed further in Sec. IV. radiation,ffiffiffiffiffiffiffiffiffiffiffiffiffi whose n dependence is small and of order Note that in order to obtain the correct Boltzmann-weight p − 2 S M ∝ En= TH 1= e bhð Þ. Note that in order to obtain the correct coefficients, e , it is crucial that the hard and − En=TH soft modes are well scrambled, so that c take values Boltzmann factor, ∝ e , it is essential that the coef- nina statistically independent of n. This construction, therefore, ficients cnina take generic values across all low-energy species, i.e., for n to run over all low-energy species [10].9 works only for a black hole stretched horizon, which does The form of the state in Eq. (11) with the statistic not have a low-energy structure. A regular material surface properties described above allows for the following coarse- does not admit an analogous construction because of the grained description of the dynamics of the hard modes. lack of this universality and hence does not have near Given the state of the system at time t [Eq. (11)], we can empty interior spacetime. The coarse graining described define a set of coarse-grained states each of which is here is the origin of the apparent uniqueness of the infalling entangled with a specific hard mode state: vacuum, despite the existence of exponentially many black hole microstates. − eSbhXðM EnÞ XeSrad We stress that the meaning of the coarse graining here is ðnÞ ⟫ kfnαg⟫ ∝ cni ajψ ijϕai; ð18Þ that a single state kfnαg in the semiclassical, effective n in — — in¼1 a¼1 theory the left-hand side of Eq. (20) corresponds to multiple different microstates—the right-hand side of where we have used the same label as the corresponding Eq. (20)—that depend on the state of the black hole and hard mode state to specify the coarse-grained state, which radiation represented by the coefficients cnina. This intro- we denote by the double ket symbol. Using Eq. (15), the duces a statistical nature in the description of the interior squared norm of the (non-normalized) state on the right- spacetime, despite the fact that the microscopic theory is a hand side of Eq. (18) is given by single unitary theory that does not have any ensemble aspect at the fundamental level. − eSbhXðM EnÞ XeSrad jc j2 nina III. ENSEMBLE FROM COARSE GRAINING in¼1 a¼1 −En In this section, we will see how the unitary gauge e TH 1 1 O pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 construction of the previous section can elucidate the origin ¼ P −Em þ − ð Þ T SbhðM EnÞ Srad ð me H Þ e e of the ensemble nature of the gravitational path integral encountered in the global gauge construction. In discussing for generic black hole and radiation microstates. Here, the this, we will ignore the existence of hard modes and the second term in the square brackets represents the size of energy constraint imposed on the hard and soft mode system, mainly because the analysis of Ref. [18], which we 9Here, low-energy species mean those below the local will discuss here, does not consider these elements. Hawking temperature. In particular, near the stretched horizon As we have seen in the previous section, the separation 1 they include all species below the string scale =ls. between the hard and soft modes as well as the energy 086021-7 KEVIN LANGHOFF and YASUNORI NOMURA PHYS. REV. D 102, 086021 (2020) constraint imposed on them are vital in understanding the A puzzling feature of this result is that while the K black emergence of the interior. These are, however, not critical in hole microstates satisfy seeing physics associated with unitarity (and hence one could get the correct Page curve without taking these hψ Ijψ Ji¼δIJ ð26Þ elements into account, as was the case in recent studies). One way to see this is to perform the Schmidt decom- by construction, Eq. (25) implies position in the space of soft mode and radiation states for each n in the state of Eq. (11): 2 −S jhψ Ijψ Jij ¼ δIJ þ Oðe bh Þð27Þ X XN n jΨðMÞi ¼ cn jH ijS ijR i; ð22Þ because in n n;in n;in n in¼1 1 XK where jH i, jS i, and jR i represent the states of the Trðρ2Þ¼ jhψ jψ ij2: ð28Þ n n;in n;in K2 I J hard modes, soft modes, and radiation, respectively, and I;J¼1 S ðM−EnÞ S N n ¼ minfe bh ;e rad g: ð23Þ If taken literally, the two equations (26) and (27) are incompatible. Reference [18] suggested that this apparent We see that the entanglement necessary for interior space- contradiction might arise because the “true” quantum time has to do with the index n, while that responsible for amplitude is, in fact, given by unitarity has to do with the summations of indices in.In S =2 fact, after the Page time (when the unitarity becomes an hψ Ijψ Ji¼δIJ þ Oðe bh ÞRIJ; ð29Þ S ðM−E Þ issue), N n is given by e bh n , so that the entanglement between the black hole and radiation comes mostly from where RIJ is a random variable with mean zero, while the the vacuum index i0 shared between the soft mode and gravitational path integral computes some type of average radiation states. In particular, hard modes play only a over RIJ so that minor role. This justifies our neglect of hard modes in the discussion 2 − hψ jψ i¼δ ; jhψ jψ ij ¼ δ þ Oðe Sbh Þ: ð30Þ in this section. We will only reinstate them and the I J IJ I J IJ energy constraint in the next section, when we discuss If Eqs. (26) and (27) are interpreted to mean Eq. (30), then reconstruction of the interior. there is no real inconsistency. The idea that the gravitational path integral may in fact A. Ensemble in the global gauge construction be computing some coarse-grained version of quantities, In Ref. [18], some of the gravitational path integrals are averaged over some microscopic information, has attracted calculated at the nonperturbative level for a simple model of much attention recently [58,71–85]. But if so, what kind of an evaporating black hole. The authors considered an average is it taking and over what ensemble? The (1 þ 1)- entangled state between the black hole (soft mode) micro- dimensional gravity theory studied in Ref. [18] is indeed states and early radiation known to be dual to an ensemble of one-dimensional quantum theories [86–88], but the concept that the dual of a 1 XK gravitational theory be an ensemble of theories seems to be jΨi¼pffiffiffiffi jψ ijIið24Þ at odds with what is known for such dualities in higher K I I¼1 dimensions [13]. Below, we see that the statistical nature of the path integral found in Ref. [18] can arise from and computed the trace of powers of the reduced density coarse graining [9–11] needed to erect an effective theory ρ matrix R of the radiation, which is the same as that of of the interior. the reduced density matrix ρ of the black hole. Here, jψ Ii 1 … K and jIi (I ¼ ; ; ) are an a priori complete set of black B. Interpretation in the unitary gauge construction hole microstates and the states of the radiation entangled with them. The result they obtained is Let us take the unitary gauge construction. Consider microstates of the form of Eq. (11), in which the black hole ( 1 Sbh is put in the semiclassical vacuum. As we have seen in N−1 for K ≪ e ρN ≈ K Sec. II B, coarse-grained states of the soft modes and early Trð Þ 1 ð25Þ Sbh −1 for K ≫ e ; eðN ÞSbh radiation corresponding to states in the effective second exterior are given by Eq. (18). For our purpose, it is where Sbh is the coarse-grained entropy of the black hole. sufficient to focus on the vacuum microstates, i.e., the states 086021-8 ENSEMBLE FROM COARSE GRAINING: RECONSTRUCTING THE … PHYS. REV. D 102, 086021 (2020) ffiffiffiffiffiffiffiffi 10 p in which the hard modes are not excited: ∀ α;nα ¼ 0. I ∼ 1 Sbh I Since we statistically expect that jdi j = e with di These states are given by having random phases, it is exponentially suppressed for I ≠ J: eXSbhðMÞ XeSrad pffiffiffi kΩ⟫ ¼ z ciajψ iijϕai; ð31Þ 1 ψ ψ ≈ ffiffiffiffiffiffiffiffi i¼1 a¼1 Ih j iJ O p : ð36Þ eSbh where we have denoted f∀ α;nα ¼ 0g by Ω and dropped ψ ψ Ω ψ This is the case even if j iI and j iJ are not orthogonal, the specification of in theP index i, soft mode states j ii, − I − J I ≪ 1 En=TH unless jðdi di Þ=di j for the majority of i (which and coefficients cia; z ¼ n e is a factor associated with the normalization. Different microstates correspond to requires a double exponential coincidence). Moreover, when we average Eq. (35) over the space of microstates different values of the coefficients cia. We introduce the index A to specify the microstate so that using the Haar measure, we get Z XeSbh XeSrad ψ ψ ψ ψ pffiffiffi Ih j iJ ¼ dUUðIÞh j iUðJÞ jΩi ¼ z cA jψ ijϕ i: ð32Þ A ia i a Z i¼1 a¼1 XeSbh UðIÞ UðJÞ 0 ¼ dU di di ¼ ð37Þ Here and below, we drop the argument M of Sbh. i¼1 We now show that the ensemble of soft mode (black hole) microstates defined by the collection of randomly if I ≠ J (barring a possible double exponentially sup- ψ 1 … K selected states j iI (I ¼ ; ; ) pressed coincidence). Here, UðIÞ represents the state obtained by acting a unitary rotation U on the state I in S S Xe bh Xe bh the space of microstates of dimension eSbh . Therefore, if ψ I ψ I 2 1 j iI ¼ di j ii; jdi j ¼ ð33Þ the gravitational path integral is indeed computing the i¼1 i¼1 average over the space of microstates, it is meaningful to “ ” ψ has precisely the feature in Eq. (30), and the reduced consider independent (overcomplete) microstates j iI 1 … K K ≫ Sbh density matrices obtained from the ensemble in Eq. (32) (I ¼ ; ; ) with e , as was done in Ref. [18]. reproduce the result in Eq. (25). This implies that, while The computation of the second quantity in Eq. (30) goes the unitary gauge construction cannot see possible null similarly. We obtain states that have already been projected out in going to the Z XeSbh construction, the ensemble remaining in it is still able to 2 UðIÞ UðJÞ UðJÞ UðIÞ jhψ jψ ij ¼ dU d d d d explain the puzzling feature of the gravitational path I J i i j j i;j¼1 integral described in Sec. III A. Specifically, the ensemble − δ Sbh nature of the gravitational path integral arises from the fact ¼ IJ þ Oðe Þ: ð38Þ that the exponentially dense spectrum of microstates caused by a large redshift makes the semiclassical path We thus find that the feature in Eq. (30) is reproduced by integral unable to resolve these microstates. In particular, the ensemble of black hole microstates in Eq. (33). the phenomenon does not require an ensemble of different Let us now consider microstates of the black hole and quantum theories. radiation system given in Eq. (32). Remember that we are We first note that the ensemble in Eq. (33) practically ignoring hard mode excitations, so the states in Eq. (32) N with the hard mode vacuum attached (which we omit here) contains the double exponential number bh;eff of “independent” states are the appropriate microstates for the system. For each microstate A, the reduced density matrix of the black hole is S N ≈ e bh ≫ Sbh given by bh;eff e e : ð34Þ S S To see this, one can compute an inner product of different Xe bh Xe rad ρ Ω Ω A A ψ ψ microstates A ¼ Trradj iAAh j¼z ciacja j iih jj: ð39Þ i;j¼1 a¼1 XeSbh ψ ψ I J Thus, Ih j iJ ¼ di di : ð35Þ i¼1 XeSbh XeSrad TrðρNÞ¼zN cA cA cA cA …cA cA : A iN a1 i1a1 i1a2 i2a2 iN−1aN iN aN 10 … The argument below, however, also applies to the collection i1;…;iN a1; ;aN of microstates in which the hard modes take an excited configuration. ð40Þ 086021-9 KEVIN LANGHOFF and YASUNORI NOMURA PHYS. REV. D 102, 086021 (2020) Note the staggered structure for the summations of i and a regardless of the relative size between eSrad and eSbh . The indices. expression in Eq. (46) can then be written as ρ To evaluate this, we can consider A as a matrix in the ( 1 Sbh ði; jÞ space −1 for K ≪ e ρN ≈ KðN Þ Trð Þ 1 ð49Þ Sbh −1 for K ≫ e ; XeSrad eðN ÞSbh ρ ≡ A A ð AÞij z ciacja ; ð41Þ a¼1 reproducing Eq. (25). We emphasize that the result here is obtained in a single which is an eSbh × eSbh Hermitian matrix of unitary theory without invoking an ensemble over different Hamiltonians at the fundamental level. The semiclassical ρ Sbh Srad gravitational path integral simply cannot resolve exponen- rankð AÞij ¼ minfe ;e g: ð42Þ tially degenerate black hole microstates and hence gives A results corresponding to the average over these states. This Using the statistical properties of the coefficients cia discussed around Eqs. (15) and (16), we find that this implies that while the semiclassical gravitational path matrix has the following elements after diagonalization: integral cannot probe details of each microstate, it knows it is dealing with a collection of states. Indeed, this is 1 consistent with the fact that the Bekenstein-Hawking ρ ≈ 0 1 … Srad ð AÞii O > ði ¼ ; ;e Þð43Þ entropy can be read off using the Euclidean path integral eSrad method [89]. for Srad ð46Þ A. Effective theory of the interior: Reconstruction without time evolution where V represents unitaries acting on the space of Suppose that the semiclassical vacuum state is given by SbhþSrad microstates of dimension e . Eq. (11) at time t. Here, t is the time measured at To connect this with the analysis of Ref. [18], we note asymptotic infinity, which we refer to as the boundary that the soft mode states entangled with different radiation ϕ time. With the coarse graining in Eq. (20), this state can be states j ai, i.e., written as Eq. (21). The question of the interior is if there is a description in which a small excitation falling into the XeSbh 1 black hole, represented by excitations of hard modes in the jψ i ≡ P cA jψ iða ¼ 1; …;eSrad Þð47Þ a A eSbh A 2 ia i zone, sees a smooth horizon and near empty spacetime j¼1 jciaj i¼1 inside it. Such a description, if it exists, cannot be the one based can all be regarded as independent even if eSrad >eSbh ,as on time evolution generated by the boundary Hamiltonian. discussed above (unless eSrad is double exponentially large, S In that description, a small object falling into the black hole Srad ≳ e bh e e ). This implies that the number of terms in is absorbed into the stretched horizon once it reaches there, the maximally entangled state of Ref. [18] should be S whose information will be later sent back to ambient space identified as e rad : by Hawking emission. To describe the object’s experience, we need a different time evolution operator associated with K ≈ Srad e ð48Þ the proper time of the object. 086021-10 ENSEMBLE FROM COARSE GRAINING: RECONSTRUCTING THE … PHYS. REV. D 102, 086021 (2020) − − X eSbhXðM En− Þ eSbhXðM EnÞ XeSrad A small object in the zone can be described by standard ffiffiffiffiffi En− þEn ˜ p 2 annihilation and creation operators acting on the hard bγ ¼ z nγ e TH 1 1 1 modes n in− ¼ jn¼ a¼ X XeSrad ffiffiffiffiffi ðn−Þ ðnÞ p × c c jψ ijϕ ihψ jhϕ j; ð57Þ bγ ¼ nγ jfnα − δαγgihfnαgj; ð50Þ n−in− a njnb in− a jn b 1 n b¼ S M−E − X e bhXð nþ Þ eSbhXðM EnÞ XeSrad X pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Enþ þEn † 2 † ˜ 1 TH bγ ¼ nγ þ 1 jfnα þ δαγgihfnαgj: ð51Þ bγ ¼ z nγ þ e n 1 1 1 n inþ ¼ jn¼ a¼ XeSrad We thus consider a future evolution of a state obtained by × c c jψ ðnþÞijϕ ihψ ðnÞjhϕ j; ð58Þ nþinþ a njnb in a jn b † 1 þ acting appropriately smoothed creation operators bγ on a b¼ Ψ P vacuum microstate j ðMÞi. We are interested in a time −E =T where z ¼ e m H , n ≡ fnα δαγg, and E are the evolution operator that makes the object’s experience m n δ manifest. energies of the hard mode states jfnα αγgi as measured ˜ † At the coarse-grained level, we can define “mirror” in the asymptotic region. Since bγ decreases the energy, this operators [40] “intermediate” operator can be defined only state depend- ently [38]. However, because of the restriction to the X ˜ pffiffiffiffiffi hard modes, the final, physically relevant operators— bγ ¼ nγ kfnα − δαγg⟫⟪fnαgk; ð52Þ † † appropriately smoothed bγ, bγ , aξ, and a operators in n ξ the physical region—can be promoted to global, state- X pffiffiffiffiffiffiffiffiffiffiffiffiffi independent (linear) operators acting throughout the ˜ † 11 bγ ¼ nγ þ 1 kfnα þ δαγg⟫⟪fnαgk; ð53Þ space of microstates [11] as in Eq. (9) ; for example, n following Eq. (9), eSbhXðMÞþSrad which can be used to form annihilation and creation † A† operators for infalling modes: aξ ¼ aξ ; ð59Þ A¼1 X α β † ζ ˜ η ˜ † A† aξ ¼ ð ξγbγ þ ξγbγ þ ξγbγ þ ξγbγ Þ; ð54Þ where an appropriate smoothing is implied. Here, aξ is γ ˜ ˜ † ˜ A ˜ A† given by Eq. (55) with bγ and bγ replaced with bγ and bγ , which are obtained by taking c ’s in Eqs. (57) and (58) to X nina A † † ˜ ˜ † be those of a specific microstate, c , and A runs over a set aξ ¼ ðβξγbγ þ αξγbγ þ ηξγbγ þ ζξγbγ Þ; ð55Þ nina γ of orthogonal microstates. Note that the operators in Eqs. (57) and (58) involve both soft mode and radiation † degrees of freedom. This construction works regardless of where bγ and bγ are the operators in Eqs. (50) and (51), ξ is ω the age of the black hole, i.e., both before and after the the label in which the frequency with respect to t is traded Page time. with the frequency Ω associated with the infalling time, α β ζ η If the black hole is younger, i.e., before the Page time, and ξγ, ξγ, ξγ, and ξγ are the Bogoliubov coefficients one can use the Petz map to obtain operators that act only calculable using the standard field theory method [69,70]. on the soft modes: The generator of the time evolution we are looking for − − would then be given by X eSbhXðM En− Þ eSbhXðM EnÞ ffiffiffiffiffi En− þEn ˜ S p 2 X bγ ¼ ze rad nγ e TH † † 1 1 Ω n in− ¼ jn¼ H ¼ aξaξ þ Hintðaξ;aξÞ: ð56Þ ξ XeSrad × c c jψ ðn−Þihψ ðnÞj; ð60Þ n−in− a njna in− jn 1 The question is if we can find microscopic operators that a¼ realize this and have appropriate properties to play the role of quantum operators in a semiclassical theory. This was studied in Ref. [11], in which such operators 11 ˜ † The noninjective nature of bγ discussed in Ref. [38] leads to were given. The simplest possibility is to use Eq. (20) in only exponentially suppressed effects in the description based on Eqs. (52) and (53) to get the global linear operators. 086021-11 KEVIN LANGHOFF and YASUNORI NOMURA PHYS. REV. D 102, 086021 (2020) S M−E − X e bhXð nþ Þ eSbhXðM EnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi Enþ þEn ˜ † S 2 bγ ¼ ze rad nγ þ 1 e TH n 1 1 inþ ¼ jn¼ XeSrad ðnþÞ ðnÞ × cn i ac jψ ihψ j: ð61Þ þ nþ njna inþ jn a¼1 We, however, cannot obtain analogous operators acting only on radiation [11].12 The existence of linear operators reproducing the correct semiclassical algebra throughout the space of microstates − (up to the intrinsic ambiguity of order e Eexc=TH ) implies that there is a sector in the microscopic theory which encodes FIG. 2. Within the spacetime region described by the effect the experience of the object in near empty spacetime after it theory of the interior (diamond at the center), the interior crosses the horizon. This allows us to erect an effective hypersurface having the maximal volume (Σ in red) is bounded theory of the interior at time t. Since the effective theory is by the codimension-2 surfaces given by the intersections of the obtained by coarse graining the region outside the zone horizon and future-directed light rays emitted from r ¼ rz and its mirror. The volume of this hypersurface is finite. (radiation), it describes only a limited spacetime region: the causal domain of the union of the zone and its mirror region on the spatial hypersurface at t in the effective two-sided B. Entanglement wedge reconstruction: geometry. Reconstruction with time evolution This limitation of the spacetime region solves the There has been significant recent progress in under- problem of an infinite volume. The maximal interior standing the interior of an evaporating black hole in the volume one can consider is now that of hypersurfaces global gauge construction [15–19]. The analyses employ bounded by the codimension-2 surfaces given by the either holographic entanglement wedge reconstruction intersections of the horizon and future-directed light rays [52–57], which builds on relations between bulk quanti- emitted from r ¼ rz and its mirror; see Fig. 2. Since the ties and boundary quantum entanglement [90–93],or state of the black hole in the effective theory is given Euclidean gravitational path integral including the effect by the thermofield double form at the time when the of replica wormholes [18,19]. According to these analyses, — effective theory is erected regardless of the age of the operators acting on early radiation are sufficient to recon- — original one-sided black hole this volume is finite. For struct a portion of the black hole interior after the Page 1 ablackholein(d þ )-dimensional flat spacetime, for time. On the other hand, we have seen that our construction ≈ d example, it is given by Vmax OðrþÞ,whererþ is the of the effective theory must involve soft mode degrees of horizon radius. The amount of entropy of semiclassical freedom in addition to early radiation. How can the relation matter one can place in this volume, without causing between the two approaches be understood in the unitary significant backreaction to the geometry, is indeed much gauge construction? smaller than the Bekenstein-Hawking entropy of the A key ingredient is the boundary time evolution. In black hole. A similar statement also applies to a large general, entanglement wedge reconstruction assumes that ≈ d−1 AdS black hole, where Vmax Oðrþ lÞ with l being the we know the time evolution operator of the boundary AdS radius. theory, in the models in AdS spacetime discussed in The fact that an effective theory represents only a limited Refs. [15–19] the Hamiltonian of a system consisting of spacetime region implies that the picture of the whole boundary conformal field theory and any auxiliary theory interior, as described by general relativity, can be obtained coupled to it. In addition, in these models it is assumed that only by using multiple effective theories erected at different the information leaked from the boundary conformal field times. In the global gauge construction, this is manifested theory—representing the bulk with a black hole—to the in the fact that seemingly independent interior states are not auxiliary system—a system storing Hawking radiation—is actually independent, as we have seen in Sec. III. effectively irreversible. These conditions allow us to reconstruct a portion of interior spacetime given the state 12The Petz map was used in Ref. [18] to construct interior of radiation at some time after the Page time, using the operators acting only on radiation. This was possible because the boundary time evolution [11]. analysis did not consider the energy constraint imposed on the Let us discuss how this works in more detail. It uses the black hole system, i.e., the hard and soft modes. Indeed, if this fact that given the complete knowledge about radiation constraint were not imposed, then the Petz map construction analogous to Eqs. (60) and (61) would work to give operators after the Page time tPage, the information that was fully acting only on radiation after the Page time [11]. scrambled into the black hole can be recovered from it once 086021-12 ENSEMBLE FROM COARSE GRAINING: RECONSTRUCTING THE … PHYS. REV. D 102, 086021 (2020) − the appropriate amount of quanta emitted after the We now take t tw to be (sufficiently) larger than the 14 scrambling is added to it [5]. Suppose we want to scrambling time tscr [5,6] : reconstruct the interior using the state of radiation at 1 time t (>tPage). Suppose also that the state of the system − ≈ t tw >tscr 2π ln Sbh: ð67Þ at some time tw (tPage En ¼ Eμ þ Eν; ð65Þ Thus, by the time t, the modes represented by ν are all emitted as radiation. We describe this as where n ¼fnαg, μ ¼fμβg, and ν ¼fνγg. With this convention, the state in Eq. (62) can be ðν;aÞ jfνγgijϕai → V jfνγgijϕai ≡ jϕν;aið72Þ written as for each ðν;aÞ. − − X X eSbhðXM Eμ EνÞ † fðfbγ gÞjΨðMÞi ¼ 14More precisely, the explicit expression of the scrambling μ ν i μ ν 1 ð ; Þ¼ time in Eq. (67) is applicable to ingoing Eddington-Finkelstein S ≈ 1 2π Xe rad time: vscr ð = THÞ ln Sbh [15,94]. The boundary scrambling ðμ;νÞ ≈ − × dμν jfμβgijfνγgijψ ijϕ i: time tscr is in general smaller: tscr vscr tsig,wheretsig is the iðμ;νÞa iðμ;νÞ a a¼1 signal propagation time between the stretched horizon and the location where information is extracted, in this case the edge of ð66Þ the zone. We ignore this subtlety here, since the difference 1 1 ≈ 1 between vscr and tscr is at most of Oð Þ: 086021-13 KEVIN LANGHOFF and YASUNORI NOMURA PHYS. REV. D 102, 086021 (2020) Substituting Eqs. (68) and (72) into Eq. (66), we find that causing the evolution in Eq. (75) by W, we can reproduce the state (of the soft and far modes) at time t becomes the effect of acting bγ on the state at tw by operators acting on the state at t: XX XeSrad 0 ðνÞ Ψ ∼ dν ψ ϕν ; † † † j it iνaj iν ij ;ai ð73Þ Bγ ¼ WVUbγU V W ; ð76Þ ν iν a¼1 † † † and likewise for bγ : ðBγ;bγÞ → ðBγ ;bγ Þ. Here, U and V are up to effects discussed below. Note that iν is a function of μ ðνÞ given by the unitaries in Eqs. (68) and (72) as U ¼⊕ν U and iðμ;νÞ [Eq. (69)], so that the summation over iν is, in fact, ðν;aÞ ðνÞ μ and V ¼⊕ ν;a V , where it is understood that U and the summations over and iðμ;νÞ: ð Þ Vðν;aÞ come with the projection operators onto the states S M−Eμ−Eν ν ν;a X X e bhðX Þ with the corresponding values of and ð Þ. Reflecting I B 0 74 the fact that rec can be fully accessed in radiation states, γ ¼ ð Þ † μ 1 B iν iðμ;νÞ¼ and γ operate only on the radiation component when acted on a state at time t. ≡ A similar construction also works for the infalling mode and we have defined dνiνa dμνi μ ν a. The actual state at t ð ; Þ operators: contains hard modes at that time, which were a part of soft or far modes at time tw. With the chaotic dynamics at the † † † stretched horizon, the population of these modes is dictated Aξ ¼ WVUaξU V W ; ð77Þ by thermality, and the state takes the standard form of † † † Eq. (11). Since the number of hard modes is much smaller and likewise for aξ: ðAξ;aξÞ → ðAξ;aξÞ. As discussed in † than that of soft modes, however, the effect of the Sec. II A, the Fock spaces obtained by acting A ’son appearance of these modes on the other sectors is minor, ξ different, orthogonal vacuum microstates are orthogonal up so we ignore it. to the exponentially suppressed ambiguity. We can there- There is one important effect which we have not yet fore form global, approximately state-independent opera- taken into account: conversion of soft modes into radiation 16 tors as in Eq. (9): outside the zone through Hawking emission. This is the effect that allows us to reconstruct the interior based only eSXbhþSrad on radiation, through the Hayden-Preskill protocol [5]. ˆ ðAÞ ν Aξ A P ; 78 ψ ð Þ ¼ ξ A ð Þ Notice that the soft mode states j iν i and radiation states A¼1 jϕν;ai in Eq. (73) contain the structure needed to reconstruct μ ν ˆ † and and the degrees of freedom entangled with them. In and similarly for A . Here, A runs over a set of orthogonal I ξ these states, let us separate the degrees of freedom, rec, I states that were vacuum microstates at tw, and PA is the needed for our reconstruction at t, and the rest, junk. The projection operator onto the Fock space built on A. The fact fact that the Hayden-Preskill protocol works implies that that complete information about the black hole vacuum the emission process can be written as state is available in radiation after the Page time allows us to take PA to act only on the radiation component. The global XX XeSrad operators in Eq. (78) can thus be defined within the space of 0 ðνÞ dν ψ ϕν iνaj iν ij ;ai radiation states at time t. ν iν a¼1 − X X X The construction described above works as long as t tw → crbcjbjψ jijϕr;j;bi; ð75Þ is larger than a time of order the scrambling time tscr; more r j b precisely, I I − ≳ − ≡ − where r and j are the indices for rec and junk, respectively, t tw vscr tsig t tw;max: ð79Þ and b represents the radiation degrees of freedom that are not directly associated with r or j. The point is that the Here, vscr is the ingoing Eddington-Finkelstein time needed degrees of freedom represented by r are now fully con- to recover the information after an object hits the stretched tained in radiation states. horizon, while tsig is the boundary time it takes for a signal This allows us to reconstruct the interior region that to propagate from the stretched horizon to the location can be described by the effective theory erected at tw where the information is extracted (see Appendix B). This on radiation at time t. Specifically, denoting the unitary implies that we can reconstruct the interior region that can be described by any effective theory erected before tw;max, 16This process occurs at the edge of the zone; see Refs. [9,95]. using operators that act on radiation at time t. This gives the 086021-14 ENSEMBLE FROM COARSE GRAINING: RECONSTRUCTING THE … PHYS. REV. D 102, 086021 (2020) sufficiently in the past within the entanglement wedge. Or, equivalently, to reconstruct such an object which is absorbed by the stretched horizon at time t0, one needs to use radiation at some time later than t0 þ OðtscrÞ. Specifically, to reconstruct an object carrying Sobj which enters the stretched horizon at time t0, one needs to use the state of radiation at time t with 8 < 1 ln S for S ≲ ln S 2πTH bh obj bh t − t0 ≳ ð80Þ : 1 S for S ≳ ln S : 2πTH obj obj bh This structure is not visible if we only compute the location of the entanglement wedge by the quantum extremization FIG. 3. Operators (Aξ, Bγ) acted on radiation states at time t procedure. which have the same effect as operators of the effective interior As can be seen in the discussion above, there are certain − theory (aξ, bγ) erected at time t tscr (or earlier). This implies that drawbacks in the entanglement wedge reconstruction E one can reconstruct the entanglement wedge rad from radiation using only radiation degrees of freedom. First, since the states at time t. The blue shaded region is the interior region reconstruction involves time evolution backward in time, the relevant to an object that is in the zone at time t and falls into the expressions for the bulk operators in terms of boundary black hole. The red arrow indicates a light signal propagating operators acting on radiation are highly complicated, and the from the stretched horizon to the edge of the zone. reconstructed operators are extremely fragile; i.e., a small deformation of the boundary operators destroys the success entanglement wedge of the radiation obtained in of the reconstruction. Furthermore, since the reconstructed Refs. [15–17,94,96–105]; see Fig. 3.17 spacetime region is that described by effective theories While the entanglement wedge represents the spacetime erected more than the scrambling time in the past, the region one can reconstruct from the radiation, the amount reconstructed operators do not describe the interior region of information one can reconstruct, or the size of code that is relevant to the fate of an object located in the zone subspace [107–109] one can erect, is not uniform through- region at the time when the state is given. In order to erect an out the entanglement wedge.18 This feature inherits effective theory that is capable of describing future evolu- from the use of the Hayden-Preskill protocol—and more tion of such an object, we need to use operators fundamentally the boundary time evolution—in the that act both on the soft modes and radiation as discussed reconstruction. Because of the scrambling and quantum in Sec. IVA. error correcting nature of the black hole dynamics, one can I arbitrarily choose which information rec to reconstruct [5]. V. CONCLUSIONS However, there is an upper bound on the amount of The information problem of black holes boils down to information one can reconstruct, coming from the fact that the tension between the unitarity of the evolution as viewed in order to reconstruct an object carrying entropy Sobj, one from a distance (e.g., the S matrix) and the existence of near has to collect OðSobjÞ Hawking quanta, which takes a time empty spacetime inside the horizon. In this paper, we have ≳ of OðSobj=THÞ.IfSobj ln Sbh, this time is longer than the promoted the idea that there are two complementary scrambling time tscr. descriptions of an evaporating black hole at the quantum This implies that an object having entropy larger level—the global gauge and unitary gauge constructions— than Oðln SbhÞ can be reconstructed only if it is located and that we can understand aspects that are mysterious in one description better by using the other description. In the global gauge construction, one begins with the 17Here, we have focused on the interior portion of the global spacetime of general relativity, so that the existence entanglement wedge. (We have also ignored possible stretching of interior spacetime is given. The challenge then is to inside the horizon.) Depending on the setup, e.g., how and where soft and hard modes are converted or extracted to radiation, the understand the unitarity of the evolution, as illustrated by entanglement wedge may contain an exterior region as in the original, “naive” calculation by Hawking [24]. This Refs. [96,99,104,106]. This can occur because the relevant far issue has recently been addressed successfully, at least in mode degrees of freedom may be able to represent hard mode simple models of gravity in low dimensions, by the operators there, e.g., with the outgoing hard modes escaping the discovery of new saddles in the gravitational path integral zone directly while ingoing hard modes first converted into soft modes (at the stretched horizon) and then escaping. [18,19]. Entanglement wedge reconstruction was employed 18A similar issue was discussed in Refs. [15,110] in a setup to understand which degrees of freedom carry information without the energy constraint. about the interior [15–17]. We note, however, that 086021-15 KEVIN LANGHOFF and YASUNORI NOMURA PHYS. REV. D 102, 086021 (2020) entanglement wedge reconstruction does not explain how APPENDIX A: STRETCHED HORIZON AND the interior emerges; the existence of the interior is SOFT MODES assumed. It simply says that if the interior exists (and it In this Appendix, we show the following: does by construction), then the degrees of freedom whose (i) The stretched horizon—the location at which the entanglement wedge contains a portion of the interior can classical description of spacetime breaks down—can be used to reconstruct it by the protocol of Ref. [5]. be defined either as a surface on which the local In the unitary gauge construction, one instead begins with Hawking temperature becomes the inverse string a manifestly unitary description. This description corre- length or a surface whose proper distance from the sponds to the picture of a black hole as viewed from a mathematical horizon is the string length; and these distance. Since the (stretched) horizon in this picture behaves two definitions agree. as a physical membrane [3], there is no problem in under- (ii) The mass and entropy of a black hole can be viewed standing unitarity; the only difference from a regular material as being carried by the soft modes—modes whose surface in this respect is that we do not (yet) know the frequencies are of order the local Hawking temper- explicit microscopic dynamics of the surface. The fact that ature or smaller. the stretched horizon behaves as a material surface, however, We focus on a spherically symmetric black hole in d þ 1 brings the question of the interior [26]; what is special about dimensions (d ≥ 3) which is not near extremal. the stretched horizon, allowing an object to fall through it The metric is given by without even noticing it? This issue was addressed in Refs. [9–11] by providing explicit constructions of operators 2 2 1 2 2 2 that make the fate of the fallen object manifest. Special ds ¼ −fðrÞdt þ dr þ r dΩ −1: ðA1Þ fðrÞ d dynamical properties of the horizon—maximally chaotic, fast scrambling, and universal—are crucial for the success of The mathematical (outer) horizon r ¼ rþ is then given by the constructions, singling out the stretched horizon. The the largest real positive root of fðrÞ: constructed operators, which include the generator of infal- ling time evolution, are linear throughout the space of fðr Þ¼0: ðA2Þ microstates and satisfy the algebra of standard quantum þ field theory in near empty spacetime, up to exponentially The temperature and entropy of the black hole are given by small ambiguities. It is therefore plausible that the infalling object indeed experiences the smooth horizon. 0 f ðrþÞ The fact that the two seemingly very different con- TH ¼ 4π ðA3Þ structions lead to the same physical conclusions implies that they are different descriptions of the same system. It is and reasonable to expect that this is a manifestation of enor- mous nonperturbative gauge redundancies of a gravita- d−1 rþ Ω tional theory discussed in Refs. [12–14]. In this context, the Sbh ¼ 4 volð d−1Þ; ðA4Þ GN appearance of a horizon seems to be indicative of the situation where the nonperturbative redundancies play an respectively, where GN is the Newton constant and important, leading role in understanding the correct phys- d=2 volðΩd−1Þ¼2π =Γðd=2Þ is the volume of the (d − 1)- ics. And as such, similar issues are also expected to arise in dimensional unit sphere. The local temperature is inflationary cosmology [10,20,21]. It is interesting to defined as understand the full implications of these nonperturbative redundancies, which would indeed have a fundamental 0 pTffiffiffiffiffiffiffiffiffiH fpðrffiffiffiffiffiffiffiffiffiþÞ importance in quantum cosmology. TlocðrÞ¼ ¼ : ðA5Þ fðrÞ 4π fðrÞ ACKNOWLEDGMENTS We are grateful to Adam Bouland, Chitraang Murdia, Pratik Rath, and Douglas Stanford for discussions on this 1. Stretched horizon and related topics. This work was supported in part by the We can define the stretched horizon r ¼ rs, using the Department of Energy, Office of Science, Office of High condition on the local temperature: Energy Physics under Contract No. DE-AC02-05CH11231 and Grant No. DE-SC0019380. 1 ≈ TlocðrsÞ 2π ; ðA6Þ ls Note added.—While completing this paper, Ref. [111] δ appeared which has some overlap with Sec. III of where ls is the string length. By writing rs ¼ rþ þ r, this paper. we obtain 086021-16 ENSEMBLE FROM COARSE GRAINING: RECONSTRUCTING THE … PHYS. REV. D 102, 086021 (2020) rffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 0 1 0 d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ðrþÞ f ðrþÞ − − ω2 χ 0 T ðr þ δrÞ¼ ¼ ðA7Þ 2 þ VlðrÞ ωLd ¼ ; ðA14Þ loc þ 4π δ 4π δ ðdr Þ fðrþ þ rÞ r at the leading order in δr, so that where r is the tortoise coordinate, dr ¼ dr=fðrÞ, and χ φ ωLd are the modes of a scalar field defined by f0ðr Þl2 δr ≡ r − r ≈ þ s : ðA8Þ 1 X s þ 4 φ Ω χ Ω −iωt ðt; r; Þ¼ d−1 ωLd ðrÞYLd ð Þe : ðA15Þ r 2 ω;Ld Alternatively, we may define the stretched horizon as a surface which is a string length away from the mathemati- The explicit form of the potential is given by cal horizon: Z d − 1 f r 2 f0 r d − 3 r ð Þ ð Þ s pdrffiffiffiffiffiffiffiffiffi ≈ VlðrÞ¼ 2 r þ ls: ðA9Þ 2 r fðrÞ 2 rþ fðrÞ l l − 2 fðrÞ þ ð þ d Þ 2 : ðA16Þ In this case, the leading-order expansion gives r Z Z pffiffiffiffiffi δ This will be analyzed later to determine rz. rs dr rþþ r dr 2 δr pffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : The results in Eqs. (A11) and (A12) show that the 0 − 0 rþ fðrÞ rþ f ðrÞðr rþÞ f ðrþÞ entropy and energy carried by the soft modes correctly ðA10Þ reproduce, parametrically, the entropy and energy of the black hole, with the latter being measured at the stretched This again leads to Eq. (A8). horizon, where most of the soft modes reside. This allows us to view that the entropy and energy of the black hole are indeed carried by the soft modes. 2. Soft modes as black hole microstates Let us integrate the entropy and local energy densities of 3. Examples the soft modes from the stretched horizon r ¼ rs to the edge of the black hole region r ¼ r . This gives We now discuss explicit examples, applying the results z so far. This elucidates some issues that were not discussed Z d−1 d d−1 d−1 d−1 explicitly, e.g., how we should choose the edge of the black rz r dr NT r Nr r ∼ d pffiffiffiffiffiffiffiffiffi ∼ H þ ∼ þ ∼ þ hole region r . S N TlocðrÞ dþ1 d−1 d−1 ; z 0 2 δ 2 l G rs fðrÞ f ðrþÞ r s N The metric of (d þ 1)-dimensional AdS Schwarzschild ≥ 3 ðA11Þ spacetime (d ) is given by Z d−2 2 d d−1 dþ1 d−1 r r r rz r dr NT r 1 − þ − þ ∼ dþ1 pffiffiffiffiffiffiffiffiffi ∼ H þ fðrÞ¼ d−2 þ 2 2 d−2 ; ðA17Þ E N TlocðrÞ dþ2 d r l l r 0 2 δ 2 rs fðrÞ f ðrþÞ r T rd−1 M where l is the AdS radius. The black hole mass is given by ∼ pHffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ ∼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðA12Þ 0 G f ðr Þδr −gttðr Þ N þ s 2 d−2 − 1 1 rþ rþ Ω M ¼ðd Þ þ 2 16π volð d−1Þ: ðA18Þ where we have assumed that the integrals are dominated at l GN r ¼ rs (which is justified for a realistic black hole) and used Eqs. (A3) and (A8); N is the number of low-energy species The temperature and entropy of the black hole are given, below the string scale, which satisfies the relation respectively, by d−1 2 − 2 2 d−1 ls drþ þðd Þl rþ ∼ G ; A13 T ;S Ω −1 : N ð Þ H ¼ 4π 2 bh ¼ 4 volð d Þ ðA19Þ N rþl GN and M in the last expression in Eq. (A12) is the mass of the black hole. a. Flat space (or small AdS) black hole The location of the edge of the zone, r ¼ rz,is determined by analyzing the effective potential VlðrÞ This case is obtained by taking the l=r → ∞ limit, appearing in the scalar equation of motion leading to 086021-17 KEVIN LANGHOFF and YASUNORI NOMURA PHYS. REV. D 102, 086021 (2020) d−2 d−2 rþ rþ APPENDIX B: SCRAMBLING TIME VS SIGNAL f r 1 − ;Md − 1 vol Ω −1 ; ð Þ¼ d−2 ¼ð Þ 16π ð d Þ PROPAGATION TIME r GN ðA20Þ In this Appendix, we show that in all cases under consideration the scrambling time is larger, or of the same − 2 d−1 order, compared with the signal propagation time. d rþ Ω TH ¼ 4π ;Sbh ¼ 4 volð d−1Þ: ðA21Þ rþ GN 1. General consideration The stretched horizon is located at As in Appendix A, we consider a spherically symmetric 1 ≥ 3 − 2 2 black hole in d þ dimensions (d ): − ≈ d ls rs rþ 4 : ðA22Þ rþ 1 2 − 2 2 2 Ω2 ds ¼ fðrÞdt þ dr þ r d d−1: ðB1Þ The effective potential in Eq. (A16) has a peak around fðrÞ − ≈ r rþ OðrþÞ, suggesting that we should take rz near this peak19: We also use the ingoing Eddington-Finkelstein coordinates, whose metric is 1 d d−2 ≈ 2 2 2 2 rz rþ: ðA23Þ − 2 Ω 2 ds ¼ fðrÞdv þ dvdr þ r d d−1; ðB2Þ We can indeed view the surface r ¼ rz as the boundary where between the near black hole and asymptotically flat regions. v ¼ t þ r ðB3Þ b. Large AdS black hole and r is the tortoise coordinate defined by For a large AdS black hole r ≫ l, þ dr dr ¼ : ðB4Þ 2 d d f r r rþ rþ ð Þ fðrÞ¼ − ;M¼ðd − 1Þ volðΩ −1Þ; 2 2 d−2 16π 2 d l l r GNl We define the signal propagation time as the time it takes ðA24Þ for a radially outgoing light ray to propagate from the stretched horizon r to the radius r at which information is d−1 s e drþ rþ T ;S Ω −1 : extracted (in the case of spontaneous Hawking emission, re H ¼ 4π 2 bh ¼ 4 volð d Þ ðA25Þ l GN is the edge of the zone rz). Here, rs and rz are defined in Appendix A. We denote the signal propagation time in The stretched horizon is at boundary time t by 2 d r ls Δ r − r ≈ þ : ðA26Þ tsig ¼ r ; ðB5Þ s þ 4 l2 where Δr is the distance between r and r in the tortoise In this case, the effective potential in Eq. (A16) mono- s e coordinate. In ingoing Eddington-Finkelstein time, this tonically increases with r, so we can take quantity is given by ≈∞ rz : ðA27Þ Δ 2 vsig ¼ tsig þ r ¼ tsig: ðB6Þ In fact, the integrals in Eqs. (A11) and (A12) converge with On the other hand, the calculation of the entanglement r → ∞ because of the AdS nature. We can, therefore, view z wedge of radiation gives us the scrambling time v the entire AdS system as the near black hole region. This is scr [15,94]. This is the length of time in the ingoing consistent with the fact that in order to make a large AdS Eddington-Finkelstein coordinates between the time at black hole evaporate, we need to couple the AdS spacetime which an object hits the stretched horizon and the earliest with an auxiliary “bath” system, as in Refs. [15,16,112]. time at which its information can be extracted at radius re. The corresponding quantity in boundary time is thus 19 We can define the zone radius rz precisely as the radius 0 0 00 0 satisfying liml→∞VlðrzÞ¼ and liml→∞VlðrzÞ < , although − Δ tscr ¼ vscr r : ðB7Þ the precise value of rz is not important anyway. 086021-18 ENSEMBLE FROM COARSE GRAINING: RECONSTRUCTING THE … PHYS. REV. D 102, 086021 (2020) From Eqs. (B6) and (B7), we obtain the expression zone using all the modes available there. The resulting expression for the scrambling time is20 t ¼ v − v þ t : ðB8Þ scr scr sig sig 1 Since t > 0, this leads to Sbh 1 scr vscr ¼ 2π ln þ Oð Þ : ðB15Þ TH cevap − − vscr vsig > tsig: ðB9Þ The coefficient c in Eq. (B15) represents the number of In this Appendix, we demonstrate that the scrambling and evap signal propagation times satisfy stronger inequality available species when the system is viewed in two dimensions through the Kaluza-Klein decomposition, ≥ vscr vsig; ðB10Þ which can be evaluated as as expected from causality. We explicitly show this up to l Xmax 1 −2 −1 fractional corrections of order =ðln SbhÞ. Here, Sbh is the c ∼ N ld ∼ N ld ; ðB16Þ ≫ 1 evap e e max entropy of the black hole, and we assume ln Sbh . Using l¼0 Eqs. (B6) and (B7), the inequality can be translated into where we have used the fact that the number of independent t ≥ t : ðB11Þ −2 scr sig states with a fixed orbital quantum number l goes as ld 1 in d þ dimensions and Ne is the number of species 1 2. General analysis available at r ¼ re in the original (d þ )-dimensional l Here we try to make as much progress as possible in a theory. The maximal value of in Eq. (B16) is determined model-independent manner. by the condition that the energy cost of angular momentum is not larger than the Hawking temperature in the original a. Signal propagation time theory as The signal propagation time vsig in the Eddington- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Finkelstein coordinates is given from Eq. (B2) as 2 l ∼ prþffiffiffiffiffiffiffiffiffiffiffiTH ∼ rþTH Z max − : ðB17Þ r fðreÞ re rþ 2 z dr vsig ¼ : ðB12Þ rs fðrÞ The scrambling time for a large AdS black hole was For a flat space or small AdS black hole, the integral in computed in Ref. [94]. The expression found there can be Eq. (B12) is dominated by the near horizon region and so written in the form21 we can approximate it as Z − 1 re rþ dx 1 r − r Sbh ≈ 2 e þ v ¼ ln þ Oð1Þ ðB18Þ vsig ¼ ln scr 2π δ f0ðr Þx 2πT δr TH cevap r þ H 1 − ≈ re rþ 1 2π ln 2 þ Oð Þ : ðB13Þ for r − r ≲ r . The scrambling time for r − r ≫ r TH ls TH e þ þ e þ þ was not calculated in Ref. [94], but we expect that it is still ∞ For a large AdS black hole, where rz ¼ , the integral in given by the fundamental expression in Eq. (B18) as long ∼ Eq. (B12) saturates above x rþ and hence as there is an available angular momentum mode, i.e., c =N ≳ Oð1Þ. (If this is not the case, our result below − − evap e 1 re rþ ≈ 1 re rþ − ≲ 2πT ln δr 2πT ln 2 for re rþ rþ would apply only to r − r ≲ r .) For large values of r ≈ H H ls TH e þ þ e vsig ≲ 1 1 rþ ≈ 1 rþ − ≳ for which cevap=Ne Oð Þ, the information extraction 2πT ln δr 2πT ln 2 for re rþ rþ H H ls TH occurs through freely propagating radiation, so we expect ðB14Þ that vscr is constant there (at the leading logarithmic level) because of the AdS nature of spacetime. at the leading logarithmic level. Here, we have l ≫ l 20 assumed s. Here and below, we assume that the spatial dimension d is not extremely large. b. Scrambling time 21We disagree with the statement in Ref. [94] that the leading term of vscr is not given by the logarithm of the entropy of the The scrambling time for a flat space (or small AdS) black horizon of the black hole. We find that it can still be written in hole was calculated in Ref. [15] under the setup that the standard form in terms of the temperature and the entropy of Hawking radiation is extracted at some radius re in the the entire horizon. 086021-19 KEVIN LANGHOFF and YASUNORI NOMURA PHYS. REV. D 102, 086021 (2020) 3. Comparison b. Large AdS black hole We now compare the scrambling time to the signal We now discuss a large AdS black hole. Properties of propagation time. All the expressions below are given at the this black hole are given in Eqs. (A24)–(A27). leading logarithmic level. The signal propagation time is given by Eq. (B14) as 8 − 2 − 2 < 1 re rþ ≈ 2l ðre rþÞl − ≲ a. Flat space (small AdS) black hole 2πT ln 0 2 dr ln 2 for re rþ rþ ≈ H f ðrþÞls þ rþls vsig 2 2 Consider a flat space (or small AdS) black hole. : 1 rþ ≈ 2l l ≳ 2πT ln 0 2 dr ln 2 for re rþ: Properties of this black hole are given in Eqs. (A20)–(A23). H f ðrþÞls þ ls The signal propagation time and the scrambling time are ðB22Þ given by Eqs. (B13) and (B15) as The scrambling time is given by Eq. (B18) with the 1 r − r 2r r ðr − r Þ consideration about the value of cevap=Ne: v ≈ ln e þ ≈ þ ln þ e þ ðB19Þ sig 2π 0 2 − 2 2 TH f ðrþÞls d ls 8 3 −1 2 − 2 r < ðd Þl ðre rþÞl ≲ þ dr ln 2 for re 2 ≈ þ rþls l and vscr 2 3 ðB23Þ : −1 2 r r ðd Þl þ ≳ þ ln 2 for re 2 : drþ ls l 1 − 1 − ≈ Sbh ≈ ðd Þrþ rþðre rþÞ vscr 2π ln − 2 ln 2 ; ðB20Þ By taking the ratio, we obtain TH cevap d ls 8 d−1 − ≲ > 2 for re rþ rþ respectively. Taking the ratio, we obtain <> − 2 2 r3 vscr d−1ln½ðre rþÞl =rþls ≳ d−1 ≲ ≲ þ ≈ 2 2 2 2 for rþ re 2 ðB24Þ v d − 1 v > lnðl =ls Þ l scr sig > 2 2 3 ≈ : ðB21Þ : lnðr =l Þ r 2 d−1 þ s ≳ d−1 ≳ þ vsig 2 2 2 2 for re 2 ; lnðl =ls Þ l ≥ 1 ≥ 3 ≥ 1 ≥ 3 We indeed find vscr=vsig for d . and hence vscr=vsig for d . [1] J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, [11] Y. Nomura, The interior of a unitarily evaporating black 2333 (1973). hole, Phys. Rev. D 102, 026001 (2020). [2] S. W. Hawking, Particle creation by black holes, Commun. [12] D. Marolf and H. 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