The Interior of a Unitarily Evaporating Black Hole

We show that operators describing the experience of an observer falling into a horizon can be constructed at the microscopic level without contradicting unitary evolution of the black hole. For a young black hole, these operators can be chosen to depend only on the state in the black hole region, while for an old black hole, they must depend on radiation emitted earlier. The difference between the two cases comes from statistics associated with the coarse-graining performed to obtain the effective theory of the interior.


Introduction
The purpose of this paper is to study how operators describing the interior of a unitarily evaporating black hole can be constructed at the microscopic level in the framework of Refs. [1,2]. This framework describes a black hole as a state in which the hard modes-the modes relevant for describing small objects around the black hole-are entangled in a generic manner with the soft modes-the degrees of freedom comprising the majority of the black hole. This entanglement is generated by a strong, chaotic dynamics near the horizon.
A black hole forms when the system-specific properties, e.g. the details of the surface of a star, are strongly redshifted due to gravity, so that the system appears to be in a vacuum state at the semiclassical level. This redshift makes the majority of the degrees of freedom unobservable for a long time, leaving only the quantities that can be specified at infinity (e.g. the mass, angular momentum, and charge) as the properties of the vacuum. These "hidden" degrees of freedom have an exponentially large density of states [3] and are distributed mostly near the horizon. While they are very soft when measured in the asymptotic region, their intrinsic dynamical scale is larger near the horizon due to gravitational blueshift, reaching the string scale at the stretched horizon [4]. The dynamics in this region is chaotic [5,6] across all low energy species, giving generic entanglement between the hidden degrees of freedom (soft modes) and the others (hard modes) that can be used to describe small excitations around the black hole. In fact, this entanglement is the origin [1] of the well-known thermality of the black hole [7].
While the strong dynamics near the stretched horizon cannot be described by a low energy quantum field theory, we expect that it is unitary, as strongly suggested by the AdS/CFT correspondence [8]. The fact that this dynamics lies outside the validity of the low energy theory allows for avoiding the conclusion of information loss [9] reached by semiclassical calculations. The picture of the black hole interior emerges after coarse-graining the degrees of freedom that cannot be physically resolved by an infalling observer, and the consistency of this picture with black hole's unitary evolution results from a specific entanglement structure between the hard modes, soft modes, and early radiation [2].
In this paper, we analyze how the operators in the coarse-grained, effective theory can be realized in the original microscopic theory. We show that the construction of these operators is not unique. In particular, we find that the operators can be written without involving an element outside the black hole region for a young black hole which is not yet maximally entangled with the early radiation. On the other hand, for an old black hole, radiation degrees of freedom must be involved, despite the fact that the degrees of freedom describing an infalling observer are not directly entangled with the early radiation. We hope that this clarifies subtle relations between the interior and unitary evolution of an evaporating black hole.
In Section 2, we describe the framework of Refs. [1,2], highlighting features relevant for our discussion. In Section 3, we present our analysis on interior operators. In Section 4, we conclude. Throughout the paper, we focus on Schwarzschild black holes in 4-dimensional asymptotically flat spacetime (or small black holes in 4-dimensional asymptotically AdS spacetime), although the restriction on specific spacetime dimensions or on non-rotating, non-charged black holes is not essential. We adopt natural units c = ̵ h = 1, and l P denotes the Planck length.

Evaporating Black Hole in a Distant Description
A key feature of the framework in Refs. [1,2] is that the thermal nature of a black hole in a distant description can be viewed as arising from entanglement between hard and soft modes of low energy quantum fields. 1 A mode of a low energy quantum field in the zone region (also called the thermal atmosphere) is classified as a hard or soft mode, depending on whether its frequency ω, as measured in the asymptotic region, is larger or smaller than ( Here, r z ≈ 3Ml 2 P , and r s is the location of the stretched horizon, given by In a distant description, the classical spacetime picture is applicable only outside the stretched horizon, and its location is determined by the condition that the proper distance from the mathematical horizon, r = 2Ml 2 P , is of order the string length, l s . While the frequencies of the soft modes are small as measured in the asymptotic region, their intrinsic dynamical scale is larger at a location deeper in the zone, due to large gravitational blueshift. In particular, it is of order the string scale near the stretched horizon, where a majority of the modes reside. (The distribution of the soft modes is given by the entropy density that goes as the cubic power of the blueshift factor 1 1 − 2Ml P 2 r.) The dynamics of the soft modes there, therefore, cannot be described by the low energy theory. 2 It is this dynamics that is responsible for unitarity of the Hawking emission process.
The quantity ∆ in Eq. (2) is naturally taken to be somewhat, e.g. by a factor of O(10), larger than the Hawking temperature Since ∆ is the inverse timescale for single Hawking emission, the uncertainty principle prevents us from specifying the energy of the black hole better than that. Below, we will assume that the energy (mass) of a black hole is determined with this maximal precision. A superposition of black holes of masses differing more than ∆ can be treated in a straightforward manner. At a given time t, the state of the entire system-with the black hole being put in the semiclassical vacuum state-is given by Excitations on a black hole background will be discussed later. In this expression, {n α }⟩ are orthonormal states of the hard modes, with n ≡ {n α } representing the set of all occupation numbers n α (≥ 0). The index α collectively denotes the species, momentum, and angular-momentum quantum numbers of a mode, and E n is the energy of the state {n α }⟩ as measured in the asymptotic region (with precision ∆). ψ Here, we have assumed that the degeneracy of hard mode states is negligible compared with that of the soft modes. This implies that i n runs over Note that with this assumption, the total entropy of the black hole is where A(M) = 16πM 2 l 4 P is the area of the black hole, reproducing the standard interpretation of the Bekenstein-Hawking entropy. The last factor φ a ⟩ in Eq. (5) represents the set of orthonormal states representing the system in the far region r > r z .
By the black hole vacuum, we mean that there is no physical excitation identifiable at the semiclassical level. This implies that any attribute a hard mode state may have is compensated by that of the corresponding soft mode states (within the precision allowed by the uncertainty principle). In particular, this implies that soft mode states associated with different hard mode states are orthogonal ⟨ψ We also take the states in the far region, φ a ⟩, to be given by those of Hawking radiation emitted earlier, i.e. from r ≈ r z to the asymptotic region before time t. S rad in Eq. (5) is then the coarsegrained entropy of this early radiation.
We take the state in Eq. (5) to be normalized We also assume that the ultraviolet dynamics near the stretched horizon is chaotic, well scrambling the black hole state [5,6]. In particular, we assume that the coefficients c nina take generic values in the spaces of the hard and soft modes. This implies that statistically where The standard thermal nature of the black hole is then obtained upon tracing out the soft modes where ρ φ,n are (n-dependent) reduced density matrices for the early radiation. In a distant description, the system of a black hole and radiation evolves unitarily with the state taking the form of Eq. (5) at each moment in time. In particular, the entanglement entropy between the black hole and radiation follows the Page curve [10], where S vN A is the von Neumann entropy of subsystem A. Throughout the history of the black hole, the number of hard modes is much smaller than that of the soft modes. (Note that we are only interested in states that do not yield significant backreaction on spacetime, which limits the number of possible hard mode states.) Furthermore, the coarse-grained entropies of the soft modes and radiation are both of order M 2 l 2 P , except for the very beginning and end of the black hole evolution. We therefore have We stress that this relation holds both before and after the Page time, at which the coarse-grained entropy of the radiation becomes approximately equal to that of the black hole. Incidentally, by performing the Schmidt decomposition in the space of soft-mode and radiation states for each n, the state in Eq. (5) can be written as where H n ⟩, S n,in ⟩, and R n,in ⟩ are states of the hard modes, soft modes, and radiation, respectively, and This expression elucidates why the entanglement argument for firewalls [11] does not apply here. The entanglement responsible for unitarity has to do with the summations of indices i n (in fact, predominantly the vacuum index i 0 ) shared between the soft-mode and radiation states, while the entanglement necessary for the interior spacetime (see below) has to do with the index n, and these two can coexist.
Let us now discuss excitations. A small object in the zone, with the characteristic size d in the angular directions much smaller than the horizon, d ≪ Ml 2 P , can be described by annihilation and creation operators acting on the hard modes In a distant description, a small object falling into the black hole is absorbed into the stretched horizon when it reaches there, whose information will be later sent back to ambient space by Hawking emission. This description, however, is not useful for addressing the question of what the falling object will actually see. Because of a large relative boost between the object and the distant reference frame, macroscopic time experienced by the object is mapped to an extremely short time for a stationary observer at the location of the object. In particular, anything the object experiences inside the horizon occurs "instantaneously" for an observer at r = r s . Understanding object's experiences, therefore, requires time evolution different from the distant one, specifically an evolution associated with the proper time of the object.

Effective Theory of the Black Hole Interior
The effective theory describing the black hole interior can be erected at each time t by coarsegraining the soft modes and radiation: the degrees of freedom that cannot be resolved by a fallen object in a timescale available to it. Suppose that the state of the system at time t (with the black hole put in the semiclassical vacuum) is given by Eq. (5) in a distant description. We can then define a set of coarse-grained states each of which is entangled with a specific hard mode state: where we have used the same label as the corresponding hard mode state to specify the coarsegrained state, which we denote by the double ket symbol. Using Eq. (11), we find that the squared norm of the (non-normalized) state in the right-hand side is given by where the second term in the square brackets represents the size of statistical fluctuations, with # representing some number that does not depend on Ml P . Therefore, the normalized coarse-grained state {n α }⟫ is given by up to a fractional correction of order 1 e #M 2 l 2 P in the overall normalization. Using Eq. (22), the state in Eq. (5) can be written in the effective theory as which takes the form of the standard thermofield double state in the two-sided black hole picture [12,13]. We emphasize that in order to obtain the correct Boltzmann-weight coefficients, ∝ e −En 2T H , it is important that the black hole has soft modes with the density of states given by e S bh (E soft ) , and that the hard and soft modes are well scrambled, giving c nina that take values statistically independent of n. This coarse-graining leads to the apparent uniqueness of the infalling vacuum, despite the existence of exponentially many black hole microstates. The annihilation and creation operators relevant for an infalling observer can be determined if the annihilation and creation operators acting on the coarse-grained states,b γ andb † γ , are defined: where b γ and b † γ are the operators in Eqs. (18, 19), ξ is the label in which the frequency ω with respect to t is traded with the frequency Ω associated with the infalling time, and α ξγ , β ξγ , ζ ξγ , and η ξγ are the Bogoliubov coefficients calculable using the standard field theory method. The generator of time evolution in the infalling description is then given by This gives the physics of a smooth horizon. The existence of the operators a ξ and a † ξ would imply that there is a subsector in the original microscopic theory encoding the experience of an object after it crosses the horizon, but how can the operatorsb γ andb † γ be constructed?
One way is simply to takeb These operators can play the role of annihilation and creation operators in the space spanned by the coarse-grained states. In particular, their matrix elements are up to corrections of order 1 e #M 2 l 2 P . It is important to notice, however, that these operators do not satisfy the exact algebra of annihilation and creation operators at the microscopic level. Indeed, which is not the identity operator for β = γ. It is only in the space of coarse-grained states that these operators obey the algebra of annihilation and creation operators: which have corrections only of order 1 e #M 2 l 2 P . Can other microscopic operators be chosen as the annihilation and creation operators in the effective theory? One might think that any operators mapping a generic microstate of {n α }⟫ (a state in Eq. (22) with generic c nina ) to those of {n α − δ αγ }⟫ and {n α + δ αγ }⟫ work asb γ andb † γ , respectively. This is, however, not the case. Since a single coarse-grained state {n α }⟫ may correspond to many microstates, the state obtained by acting such an operator, e.g.b † γ , to the specific microstate, i.e. the state {n α }⟫ obtained using the specific c nina appearing in the state of the system in Eq. (5), may not have an appropriate inner product with the corresponding microstate, e.g. {n α + δ αγ }⟫ obtained using the c nina in Eq. (5).
As an example, consider the set of candidate operators where c is a normalization constant, n ± ≡ {n α ± δ αγ }, and with generic coefficients satisfying ∑ in f nin 2 = ∑ in g nin 2 = 1. This gives and there is no choice of c that can make both of these relations compatible with the algebra in the effective theory.
The consideration above provides an argument for the necessity of the dependence [14] ofb γ andb † γ on the state of the system, in particular c nina in Eq. (5). This, however, still allows for operators other than those in Eqs. (27, 28).
Let us consider the operators where c is a real number, and E n± are the energies of the hard mode states {n α ±δ αγ }⟩ as measured in the asymptotic region. Note that the combinations of c njna 's appearing here, ∑ so that they can be determined purely from the state in the black hole region. With this choice ofb γ andb † γ , we obtain and up to corrections of order 1 e #M 2 l 2 P . We thus find that for is dominated by the a = b terms if and only if the condition in Eq. (45) is met, giving the state proportional to and hence to the {λ α ±δ αγ }⟫ obtained using the specific c njna 's appearing in the state of the system. This shows how the Page time can be relevant in the construction of the interior operators, despite the fact that the hard mode and radiation states take separable forms as in Eq. (13) throughout the history of the black hole. In fact, for an old black hole, we do not see how one can construct the appropriate annihilation and creation operators in the effective theory using only the information in the black hole reduced density matrix in Eq. (40). Incidentally, a construction ofb γ andb † γ that involve only radiation states is not possible. What allowed the construction of operators in Eqs. (38, 39) is the correlations between the attributes of the hard and soft modes coming from the constraints imposed on the black hole vacuum state (the requirement that it does not have any features associated with semiclassical excitations). Such correlations do not exist between the hard modes and radiation.
As discussed in Ref. [1], the effective theory of the interior erected as above 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 (the time at which the effective theory is erected). The black hole singularity may be regarded as a manifestation of the fact that this theory is obtained by coarsegraining and hence represents a finite-dimensional, non-unitary system. Specific operators used in Eqs. (24, 25), for example those in Eqs. (38, 39), are selected presumably because they correspond to observables which classicalize in such a finite-dimensional system [2]. Locality seems to play a key role in this quantum-to-classical transition.
The fact that an effective theory represents only a limited spacetime region implies that the picture of the whole interior, as described by general relativity, can be obtained only by using multiple effective theories erected at different times. This is the sense in which the global spacetime in general relativity emerges from the microscopic description.

Conclusions
In this paper, we have shown that operators describing the experience of an observer falling into a horizon can be constructed without contradicting the unitary evolution of the black hole. The choice of these operators at the microscopic level is not unique. In particular, for a young black hole, we can choose them to depend only on the microscopic information in the black hole region. On the other hand, for an old black hole, the operators must depend on radiation emitted earlier.
The difference between the two cases comes from statistics associated with the coarse-graining performed to obtain the effective theory of the interior. We hope that this analysis resolves confusion surrounding Ref. [11], at least for small excitations around a black hole evaporating in asymptotically flat spacetime (or a small black hole in asymptotically AdS spacetime).
As discussed in Ref. [2], the fact that the smooth interior requires chaotic dynamics at the stretched horizon represents an intriguing relation between ultraviolet and infrared physics. This has a number of implications, including the fact that a global symmetry must be broken at the string scale by an O(1) amount. Phenomenologically, this suggests that the QCD axion has a string origin or the relevant Peccei-Quinn symmetry arises as an approximate accidental symmetry at low energies, resulting from the choice of matter representations under gauge symmetry.
While the coherence and consistency of the picture presented here appear convincing to the author, it is desirable to show elements of it in an explicit model based on holography. In particular, it would constitute great progress if some of these elements are demonstrated for small black holes in the AdS/CFT correspondence.

Note Added in v2
After the submission of this paper, 3 an interesting paper appeared [15] which discusses related issues. We claim that their models correspond to the situation in a large AdS black hole. For a large AdS black hole, it makes sense to consider an excitation that has a frequency smaller than Hawking temperature and falls into the black hole, since the inverse Hawking temperature is (much) smaller than the horizon radius, 1 T H ≪ r + . Here, T H = 3r + 4πl 2 is the Hawking temperature, and r + and l are the horizon and AdS radii, respectively. (For a small AdS black hole or a black hole in asymptotically flat space, 1 T H ∼ r + , so an excitation whose frequency is much smaller than T H does not fall into the black hole.) In this situation, states of the "relatively hard" modes (1 r + ≪ ω ≲ T H ) describing an infalling object need not be correlated with the states of the other, "relatively soft" modes (ω ≲ 1 r + ) as in Eq. (5), since the uncertainty in energy (∼ T H ) is larger than the frequencies of the harder modes. In other words, the orthogonality condition in Eq. (9) should not be imposed, and the state of the system with the black hole put in the semiclassical vacuum can be written as where e S code (≪ e S bh (M ) ) is the dimension of the Hilbert space for the harder modes, S bh (M) = π(2l 2 M l P ) 2 3 is the density of states for the softer modes (≈ that of the black hole), and φ a ⟩ represents states of the auxiliary system to which the AdS system is coupled at the boundary. 4 We can now define the normalized mirror states The statistical errors for the normalizations are fractionally of order 1 √ e S bh (M ) e S rad . Below, we focus on the regime S rad ∼ S bh (M) ≡ S bh , since we are interested in the Page transition, and denote errors of order 1 e #S bh +# ′ S rad simply by 1 e Ssys . With this setup, we find that the softer mode states that are entangled with different harder mode states are nearly orthogonal due to the large dimensionality of the relevant Hilbert space: The point, however, is that while these overlaps are small, of order 1 e Ssys , they are nonzero. This is in contrast to the case of a flat space, or small AdS, black hole in which the overlaps were virtually zero because of Eq. (9). We now see that this seemingly negligible difference allows, after the Page time, for choosing microscopic interior operators that act purely on the auxiliary system, i.e. early radiation, as discussed in Ref. [15]. The analysis is very much analogous to that in Section 3. Let us consider the operators acting only on radiation statesb This leads to and where we have omitted contributions that are guaranteed to be suppressed by 1 e Ssys . In order for the operators in Eqs. (52, 53) to be the annihilation and creation operators in the effective theory of the interior, the second terms in the square brackets in Eqs. (56, 57) must be negligible, which is the case if and only if e S bh ≪ e S rad , i.e. the black hole is old. With Eq. (60), the algebra of annihilation and creation operators, Eqs. (32, 33), is satisfied in the effective theory. It is easy to see that for a young black hole, e S bh ≫ e S rad , one can construct valid annihilation and creation operators that act only on the black hole softer modes: These operators, however, lose their validity when the black hole becomes old. The operators analogous to those in Eqs. (27, 28), which act both on the black hole softer modes and radiation, can be used throughout the history of the black hole, i.e. regardless of its age. The analysis in this paper and the present note indicates the following: • In Ref. [15], the statistical nature of bulk observables has arisen from averaging over the ensemble of boundary theories dual to Jackiw-Teitelboim gravity. We have seen that essentially the same physics arises from coarse-graining performed to obtain an effective, non-unitary theory of the interior. Allowing for errors/ambiguities that are exponentially suppressed in the system size, an operator in the effective theory corresponds to exponentially many sets of operators and states in the microscopic, unitary theory. Statistics associated with this correspondence is the origin of the statistical nature seen in the effective, intrinsically semiclassical theory of the interior.
• The statistical nature of the effective theory seen here disappears exponentially in the system size as the system becomes large, since the relevant variances decay as e −Ssys . This occurs, in particular, in the Rindler limit of black hole spacetimes, which can be taken smoothly with the entropy density per unit transversal area fixed [2], explaining why similar phenomena are not seen in theories in asymptotically flat spacetimes. This limit, however, is not available in (1 + 1)-dimensional theories of gravity. This seems to be the reason for why the dual of the Jackiw-Teitelboim gravity involves an ensemble of theories.
• In a large AdS black hole, operators describing the interior can be constructed after the Page time in a way that they act only on radiation degrees of freedom, while this is not the case in a small AdS, or flat space, black hole. The difference comes from the way in which the energy constraint is imposed on various degrees of freedom, which in turn is a consequence of different relative sizes between the Hawking temperature and the frequencies of modes describing an infalling object. This seems to suggest that the phenomenon of entanglement islands as originally discussed in Refs. [16,17] is a specific feature to a large AdS black hole and systems similar to it.