Microscopic origin of the entropy of astrophysical black holes

We construct an inﬁnite family of geometric microstates for black holes forming from collapse of dust shells in Minkowski spacetime. Quantum mechanical wormholes cause these states to have exponentially small, but universal, overlaps. We show that these overlaps imply that the microstates span a Hilbert space of log dimension equal to the event horizon area divided by four times the Newton constant, explaining the microscopic origin of the Bekenstein-Hawking black hole entropy


INTRODUCTION
Bekenstein and Hawking [1,2] proposed, on the basis of general relativity and quantum mechanics in curved spacetimes, that black holes behave as thermodynamic objects, and carry an entropy S = A/4G, where A is the area of the event horizon, G is Newton's constant, and we are working in units where Planck's constant and the speed of light are 1.This remarkable formula is universal.It applies to any black hole regardless of its mass, charge, or angular momentum, and in any spacetime dimension.
What is the origin of this entropy?Statistical mechanics asserts that the thermodynamic entropy of a classical system equals the logarithm of the number of microstates consistent with the macroscopic parameters.Quantum mechanics complicates matters.Quantum states form a Hilbert space; so any suitably normalized linear combination of microstates is also a microstate.Thus, in quantum systems we instead identify entropy as the logarithm of the dimension of the Hilbert space.To give a statistical mechanical interpretation of black hole entropy, we have to determine the dimension of the underlying quantum gravity Hilbert space describing a black hole.This fundamental problem was solved in a special case by Strominger and Vafa [3] who explained the entropy of certain supersymmetric black holes in terms of the Hilbert space of underlying string theoretic microstates.These calculations were possible because the black holes in question: (1) have multiple types of electric and magnetic charges (unlike our world where there is one electromagnetic field, and no magnetic charge); (2) are extremal (unlike most astrophysical black holes) so that, in terms of the charges, the mass achieves a certain lower bound required for avoiding naked spacetime singularities; and (3) are supersymmetric, in that they retain a fraction of the supersymmetry of the theories in which they are defined (unlike real black holes which have no supersymmetry to break).These considerations were central to the computability of the entropy.Furthermore, the analysis relied on technical details of the ultraviolet completion of gravity in string theory, which include many extra dimensions and exotic extended solitonic objects of cosmic scale.The fundamental question has thus remained: can we give a universal microscopic explanation for the entropy of astrophysical black holes?Here, we propose an answer to this question.
Briefly, we will exploit the fact that any linear superposition of quantum states is also a state.Thus, rather than a specific basis of typical black hole microstates, we simply seek any set of states that is large enough to span the entire Hilbert space.We also require this set to be under sufficient control for us to compute the Gram matrix of state overlaps.The rank of the Gram matrix determines the maximum number of linearly independent microstates, giving the dimension of the Hilbert space.Equivalently, the logarithm of this rank quantifies the thermodynamic entropy.
To this end, we construct an infinite family of atypical, but well-controlled, geometrical microstates which account for the entropy black holes forming from dust shells in Minkowski space, like in stellar collapse.Our construction follows in general relativity, and does not require any exotic ingredients.Astrophysical black holes generally have some angular momentum, but we will focus on non-rotating black holes for analytical simplicity.Extending methods developed in [4] for universes with a negative cosmological constant, we compute the overlaps of our microstates in quantum gravity.We find that they span a Hilbert space of dimension precisely equal to the exponential of the Bekenstein-Hawking entropy.This finding explains the microscopic origin of black hole thermodynamics.

BLACK HOLE MICROSTATES
We start by constructing an infinite family of internal states for an eternal, asymptotically flat (Minkowski), one-sided black hole (Fig. 1).Such black holes do not form from collapse -rather, they exist forever, and behind the event horizon there is a "white hole" singularity arXiv:2212.08623v2[hep-th] 5 Feb 2023 where time begins, in addition to a black hole singularity where time ends.However, as we will discuss below, their geometry matches the late-time behavior of black holes forming from collapse, and they can account for the entropy associated to such spacetimes.
Outside the horizon, all the states we construct match the geometry of a Schwarzschild black hole of radius r s = 2GM , where M is the ADM mass.As such, they are microstates of the same black hole as seen by an external observer, and are distinguished by their interior geometries: each contains a different configuration of matter which backreacts to generate a distinct interior.The matter emerges out of the past singularity and dives into the future singularity, without leaving the black hole region (Fig. 1).For simplicity, we restrict to matter organized in spherical thin shells of dust particles, with total rest mass m.The states in the family are labelled by the mass m of the shell in the interior.Typical black hole microstates are expected to contain Planck scale structures and other features making a geometric description difficult.By contrast, our atypical microstates have a well-defined interior: at any time the geometry of space behind the horizon caps off smoothly (Fig. 2).Previously, such"bag of gold" geometries, originally described by Wheeler [5], were thought to present a conceptual problem because they naively overcount the microstates of black holes.We will see how these atypical bags of gold can in fact account precisely for the black hole entropy.
FIG. 1. Penrose diagram of a microstate of an eternal onesided black hole.The exterior geometry extends between the future/past horizons H ± and the conformal null boundaries J ± .The interior contains a thin shell W, which divides the geometry between a region of flat space < inside the shell, and a region of black hole geometry > outside the shell.The zigazag lines at the bottom and top are the the white hole and black hole singularities where time starts and ends.
The exterior metric is of the Schwarzschild form: where f (r) = 1 − rs r and dΩ 2 = dθ 2 + sin 2 θ dϕ 2 is the round metric of the unit sphere S 2 .This metric can be continued into the interior of the black hole, at r < r s ; and further to a second asymptotic region, described by the same metric (1).Our microstates consist of a portion of this extended geometry separated by a moving shell of matter from a patch of flat spacetime that smoothly caps off the geometry without reaching a second asymptotic boundary (Figs.1,2).In the thin-shell limit, the full geometry inside and outside the shell is determined by the Israel junction conditions [6] which fixes the change in the spacetime metric across the shell.Concretely, the worldvolume W of the thin shell carries the localized energymomentum of a pressureless perfect fluid where σ is the surface density of the fluid, and u µ is the four-velocity field of the dust, tangent to W. The induced metric on the worldvolume W is determined by R(T ), the radius of the shell R as a function of its proper time T .The proper mass of the shell is constant along W and it is given by m = 4πσR 2 .Interestingly, because we are considering shells that always lie behind the horizon, the proper shell mass m can actually exceed the black hole mass M and still give a consistent solution.
From the point of view of the metric (1), the shell will live at r = R(T ), with T determined by the proper time along the shell's trajectory, and where r is the radial direction in the extended spacetime.The equation of motion for R(T ), determined by the Israel junction conditions, is that of a non-relativistic particle of zero total energy where we defined the effective potential Provided that m ≥ M , the shell will expand from the past singularity, located at finite proper time in the past.The radius of shell can be defined in a coordinate invariant manner, and grows until it reaches R * > r s at which V eff (R * ) = 0.It may seem surprising that the black hole interior can contain a sphere larger than the horizon, but this can occur because of the backreaction of the shell stress tensor.After this turning point the shell contracts agains as it dives into he future singularity.For large proper mass, m M , the shell re-collapses at a radius R * ≈ Gm/2.
The geometry inside the shell consists of a portion of flat space which caps off smoothly at r = 0 while the shell sits at r = R(T ).To better understand this geometry, we can focus on the time-reflection symmetric hypersurface Σ (Fig. 1) on which we can define the microstate via Euclidean path integral methods.When the shell mass is large m M , the total interior volume of Σ in ⊂ Σ scales as Vol(Σ in ) ≈ π 3 (Gm) 3 , while the surface of the shell σ = Σ ∩ W has maximum area scaling as Area(σ) ≈ π(Gm) 2 (see Fig. 2).The states we have constructed are labeled by the mass of the interior shell.The euclidean construction of these states is depicted in Fig. 3. Notice that the euclidean "preparation temperatures", βm , β m in Fig. 3, are not the same as the black hole temperature which is determined by the periodicity around the entire boundary.The preparation temperature βm depends on the mass of the shell and the black hole temperature via the equations of motion of the shell's trajectory, see Ref. [4] for details.The preparation temperature β m is unrestricted; different choices represent different states.

H Exterior
Finally, simple generalizations of these states follow by considering multiple shells in the interior.We could also consider other configurations of matter, including exotic matter arising from string theory.These choices will generate different interior configurations, while keeping the exterior fixed.As we will argue below, none of these details will matter for counting the microstates.

QUANTUM GRAVITATIONAL OVERLAPS
For each state we construct, the spacetime geometry X can be analytically continued to the Euclidean section along the time reflection-symmetric hypersurface Σ.The continuation defines a set of asymptotic boundary conditions at Euclidean spatial infinity ∂X.These boundary conditions can be used in a standard path integral construction to prepare quantum mechanical microstates.Related constructions have been studied in models of 2d gravity [7,8] and in AdS/CFT [4,9].In this way, the states that we have discussed in the previous section specify an infinite family of quantum states {|Ψ m } of the Hilbert space of the black hole, where m labels the proper mass of the matter inserted in the black hole interior.Note that there is no upper bound on m.This infinite family naively overcounts the Bekenstein-Hawking entropy.But this is only so if the states are orthogonal to each other.To get a correct counting of the dimension of the Hilbert space, we must compute the overlaps Ψ m |Ψ m between our states.This can be done using the semiclassical gravitational path integral.The rules are: (1) fix the asymptotic boundary conditions that prepare the respective states, and (2) fill in the Euclidean geometry with all possible saddlepoint manifolds that respect these boundary conditions.We will work with a simple effective description in terms of the Euclidean gravitational action coupled to a thin shell Here R is the Ricci scalar of the Euclidean manifold X, K is the extrinsic curvature of its boundary ∂X, σ is the density of the shell, W is the worldvolume of the shell, and I ct are counterterms, localized at the asymptotic boundary ∂X, that make the action well-defined.
The leading contribution to the overlap comes from Euclidean shell geometries.When m = m these are straightforward to construct and are simple analogs of those constructed in [4,10].When m = m , we need some way to join the different shells from the Euclidean boundary.Achieving this will require a parametrically large number of interactions, and so we expect the result to be exponentially suppressed in the shell mass differ-ence.Thus, the leading contribution will be where the overbar denotes that the calculation is performed according to the rules of the gravitational path integral, and where we have implicitly normalized the states using the on-shell action of the corresponding euclidean manifold . Therefore, microstates representing different classical geometries appear in this approximation to be orthogonal, suggesting that our family of states spans an infinite dimensional Hilbert space.
However, this conclusion is drastically changed by the appearance of semiclassical wormhole contributions in the Euclidean path integral that compute higher moments of this type of amplitude.These wormholes correspond to non-perturbative effects in quantum gravity.To start with, a two boundary wormhole contributes to the square of the overlap The new contribution is , where X 2 (m, m ) is the Euclidean wormhole manifold that extends between the two asymptotic boundaries that prepare each of the overlaps (Fig. 4).The wormhole arises by gluing a euclidean black hole geometry to flat space across two different shells.These are classical solutions to the equations of general relativity, and therefore contribute to the semiclassical amplitude (8).Similar wormhole solutions were described in [9,10] for the case of AdS space, and we are extending them here to Minkowski spacetime.The action of the wormhole can be computed straightforwardly from (6) (details in the Appendix).
Similarly, the connected contribution to the n-th product from n-boundary wormholes is (9) Here Z n = e −I [Xn] where I[X n ] is the action of a semiclassical wormhole X n with n boundaries.Again, this wormhole is a classical solution to equations of motion.As explained in the Appendix, the wormhole consists of an Euclidean black hole joined to a region of flat space through the trajectory of mutiple shells (we are here adapting a construction in AdS space from [4,10]).
These wormhole contributions induce a "nonfactorization" of the moments of the inner products: (9) is not a product of factors of the form (7). At first sight this seems disturbing.However, there is a simple microscopic interpretation of this non-factorization of overlaps.As discussed in the Appendix, the nonfactorization follows from an assumption of dynamical chaos in the black hole in the form of the Eigenstate Thermalization Hypothesis [11,12].Briefly, we expect that the semiclassical Euclidean saddlepoints do not have access to the fine-grained structure of the underlying quantum states, and in fact are computing averages over microstates that differ by tiny details that are semiclassically invisible.These states will differ in their erratic phases relative to, say, the energy eigenbasis.As such the gravitational computation is in fact averaging over the statistics of these phases leading to cancellation in the overlap (7) [4,10] but not in the higher moments of the overlap.This sort of phenomenon was first described in two-dimensional gravity by [13], but interpreted differently in terms of an ensemble of fundamental theories.
As we will see, we will only need the moments (9) in the limit of large masses m i M .As illustrated in the Appendix for the second moment, the wormhole action simplifies and reduces in this limit to independently of the shell masses.Here Z bh (β) = e −I bh (β) and I bh (β) = βM − S is the gravitational action of the Euclidean Schwarzschild black hole of inverse temperature β = 8πGM with S = A/4G.We are here interpreting the euclidean gravity action for the black hole first computed by Gibbons and Hawking [14] as the amplitude of microstate overlaps in the large mass limit.
Hence, we conclude that the overlaps become universal, and independent of the masses of the shells characterizing the microstates, when these masses are large.As we will see in the next section, these universal overlaps encode the dimension of the Hilbert space of the black hole.

COUNTING MICROSTATES
We now consider an infinite subfamily of black hole microstates { Ψ mj } for shells with mass m j = j m for j = 1, 2, ..., where m is sufficiently large to ensure that zero-wormhole off-diagonal overlaps are subleading.We seek the dimension of the Hilbert space spanned by these microstates.To this end we consider the Gram matrix G of overlaps for a finite subset of these states.This is defined by where i, j = 1, ..., Ω.The Gram matrix is hermitian and positive semidefinite, and its rank, i.e., the number of nonzero eigenvalues, gives the dimension of the Hilbert space spanned by set of states.We will compute the rank of the Gram matrix as a function of Ω.
In order to count black hole microstates for a given energy from these overlaps, i.e., the microcanonical degeneracy, we notice first that any entry of the Gram matrix is a sum over microcanonical windows in the energy basis because we constructed microstates with fixed temperature, rather than fixed mass.Thus, to compute the microcanonical Gram matrix, we need to the project the states into the microcanonical window of energy E. To do so, we need the moments of the inner product (10) projected onto each energy band.We can achieve this projection via an inverse Laplace transform [4] Z bh (nβ) = dE z(E) e −nβE . ( which then gives the microcanonical moment In the Appendix we show how to use these moments to obtain the eigenvalue density of the Gram matrix.For a given Ω the result is where S = A/4G.Ultimately the value A/4G arises from evaluation of the gravitational action of the wormhole contribution to the overlap moments.This eigenvalue density has a continuous part with support on positive eigenvalues.The singular part counts the number of zero eigenvalues.Therefore, the rank of the Gram matrix is the number of eigenvalues contained in the continuous part of the distribution.We thus conclude: • For Ω < e S , the rank of G is given by Ω since the second term in ( 14) vanishes.
• For Ω > e S , the rank of G is given by e S because the second term in (14) shows that the remain eigenvalues vanish.
For Ω < e S , we can thus use the Gram-Schmidt procedure to construct an orthonormal set of vectors.For Ω > e S this procedure will produce only e S linearly independent vectors.This is the main result of this article, namely that the black hole microstate degeneracy, equal to the number of possible orthogonal states in a given energy band is equal to the exponential of the Bekenstein-Hawking entropy.Equivalently, our construction provides a statistical understanding of the entropy of black holes in Minkowski spacetime.These statements are valid on average in the effective random Gram matrix ensemble provided by semiclassical gravity.This ensemble arises because the semiclassical Euclidean saddlepoints that we used to compute overlaps do not have access to the fine-grained phase structure of the underlying quantum states, and effectively average over an ensemble of microstates which share the same semiclassical description.Deviations from the average Gram matrix should be suppressed by the dimension, in this case by factors of e −S , as expected from the general theory of random matrices [15].It would be interesting to compute these subleading corrections to the entropy.However, they will also likely be subleading to other corrections, such as those following from higher-derivative corrections to the gravitational effective action that will generically appear in any UV-complete theory.
To summarize, even if we keep adding potential microstates, there is a point at which these states cannot be orthogonal anymore.This point is controlled by the universal statistics of the inner product, which in turn encodes the Bekenstein-Hawking entropy.As proposed in the introduction, the solution to the problem of the microscopic origin of the entropy of general black holes is not to construct a specific of e S microstates.Indeed, there are infinite numbers of such sets, even when they are constrained to be semiclassical and geometrical.The problem is to count how many orthogonal states we can build out of those, and to prove that this counting gives rise to the right Bekenstein-Hawking dimension.
Finally, one could ask if any black hole microstate is expandable in our basis.Indeed, we expect that most states can be expanded in our shell state basis.There is an explicit procedure to check this using our techniques.We must compute the inner products between the desired states and our basis, taking into account the appearance of wormholes.For example, we could take the single sided black hole state without any shell, along with the small excitations around this background.These states again have non-zero, exponentially suppressed, overlaps with all shell states that we have considered.The dominant contribution to these overlaps comes from a wormhole saddle with one shell.Thus adding the no-shell family of states to the Gram matrix will not increase its rank.

DISCUSSION: BLACK HOLES FROM COLLAPSE
Gravitational collapse to form black holes can be modeled by a spherical shell of dust falling in from infinity (Fig. 5, left).The shell, with flat space inside and a Schwarzschild geometry outside, eventually passes behind the spacetime horizon that its own presence generates.After this passage, classical observers outside the horizon observe a black hole geometry of a fixed mass M in an asymptotically flat universe, but do not know the precise state behind the horizon.In particular, all the microstates described above for the eternal black hole lead to same geometry outside the horizon as the late time shell state.Thus, by the rules of the microcanonical ensemble, the exterior classical observer will interact with the black hole as if it has an entropy equal to that of an eternal black hole of the same mass.
Most of the quantum states counted by this entropy have pasts that do not resemble the infalling shell (Fig. 5, right).For example, the interior shells that we used to count the microstates in the previous sections timeevolved from a white hole singularity of the eternal onesided black hole.Notice though that quantum states are defined on a time-slice, for example Σ in Fig. 1, and do not by themselves contain information about time evolution; all states compatible with the macroscopic description of the system contribute to its entropy, regardless of their past history.
FIG. 5. On the left, the spacetime diagram of a collapsing shell which creates a black hole and defines a state |Ψ on the spacelike hypersurface Σ.The shell is the red line emerging from past inifinity to fall behind its own horizon and eventually into the future singularity (jagged line).The black hole horizon is the diagonal black line bounding the region inside the black hole that is not in causal contact with the asymptotic boundary.On the right, a small perturbation of the state on Σ (purple line) which generates a singularity when time-evolved into the past.
The fact that we can define quantum states on a timeslice without reference to the past suggests that we can write the collapsing shell geometry at late times in the basis of interior shell states we have introduced.The reason is a generalization of arguments given above.The infalling shell state should have exponentially small but still non-vanishing overlaps with all states in the basis.These overlaps can be computed using the gravitational path integral, and will arise through a wormhole contribution similar to the ones we have been discussing.However, because of the time asymmetry of the collapsing shell, numerical analysis is required for finding the wormhole solution and its action.We leave this for future work.Conceptually, we are describing a scenario where quantum superpositions of macroscopic, semiclassical states can yield other macroscopic, semiclassical states (see [4] for a similar phenomenon concerning Einstein-Rosen bridges of different lengths).This can be consistent, because individual overlaps are extremely small and we are making use of a very high-dimensional Hilbert space together with a delicate balance of phases.

ACKNOWLEDGMENTS
We would like to thank José Barbón, Horacio Casini and Roberto Emparan for useful discussions.VB and JM are supported in part by the Department of Energy through DE-SC0013528 and QuantISED DE-SC0020360, as well as by the Simons Foundation through the It From Qubit Collaboration (Grant No. 38559).AL and MS are supported in part by the Department of Energy through DE-SC0009986 and QuantISED DE-SC0020360.This preprint is assignated the code BRX-TH-6713.

Euclidean wormhole contribution
The Euclidean wormhole solution that we consider in this paper is a modification of the solutions in [4,9,10].In contrast to the earlier work, our spacetime is locally flat.Also, we are constructing a "one-sided wormhole" which has a black hole region glued to a region of euclidean flat space along the trajectories of two thin shells, as in Fig. 6.Essentially all of the results below can be obtained from the analysis in [4] by setting the cosmological constant to zero and setting the metric in the appropriate shell-delimited region to be flat space.In view of this, we will be brief below, and refer the reader to [4] for detailed steps of the calculations.
Using the same analysis as in [4], in the large mass limit, m, m M , the solution to the shell equation of motion shows that the trajectory becomes very short in Euclidean time.Thus the portion of the Euclidean black hole boundary that is excised by the shell in Fig. 4 becomes small, and the the "preparation temperatures", FIG. 6. Euclidean wormhole contribution to the second moment of the overlap.The left part of the wormhole is an Euclidean black hole of mass M2, while the right part is a region of Euclidean flat space, with a non-contractible thermal circle.The red and orange lines represent the two shells in the wormhole in Fig. 4, cut open here along the red shell.As described in the main text and [4] the boundary "preparation times" ( βm i , βm i ) are fixed by the preparation of the states with shells of mass mi and m i respectively.
i.e., the Euclidean coordinate time of the remaining part of the spacetime boundary becomes βmi , βm i ≈ β.In this limit, the red and organge shell trajectories in Fig. 6 become very short in Euclidean time, and pinch off.At the same time the upper and lower boundaries of the region labeled M 2 in Fig. 6 will each have a Euclidean coordinate length equal to β.Thus the wormhole pinches and the mass of the black hole part of the wormhole geometry becomes M 2 ≈ 2M , associated to a Schwarzschild black hole of inverse temperature 2β in four dimensional, asymptotically flat space.This follows directly from Israel junction conditions, see [4] for details.
Therefore, in the limit of large masses, the total contribution from the wormhole Z 2 can be separated in three parts The first factor corresponds to the gravitational action of the Euclidean black hole in the left part of the wormhole in Fig. 6 Z bh (2β).The second factor corresponds to the gravitational action of the flat space part of the wormhole on the right of the orange shell in Fig. 6, Z vac ( β m,m ), for the Euclidean time β m,m = β m + β m .The last two factors are the actions intrinsic to the shells, denoted Z m and Z m respectively.The Z 2 factorizes in this way because the action is a sum of gravitational and shell contributions.
Similarly, in this regime, the gravitational action of the Euclidean manifold of Fig. 3, wich computes the normalization of the state, is given by The total wormhole contribution in ( 8) is then To arrive at this result we used the fact that the Euclidean action of flat space is extensive in Euclidean time, so that the exponential of the action satisfies Likewise, the intrinsic contributions of the shells in this limit become insensitive to the actual background manifold in which they propagate because the trajectories are short, and thus localized close to the spacetime boundary where the metric is flat.Thus they cancel between numerator and denominator.The details of this cancellation are entirely parallel to the analysis in [4].The analysis of the wormhole contributions to the n-th moment of the overlaps proceeds similarly and leads to (10) in the main text.

Microscopic interpretation of the overlaps
Inner products between quantum states in a Hilbert space are by definition complex numbers.As such, the product of a collection of them must factorize.However, this is not the outcome of the gravitational path integral calculation of the ovelaps.Indeed, the wormhole contributions in (9) manifestly spoil factorization.
This naturally raises a question: what are these overlaps really computing in the fine-grained description of the black hole?In this appendix, we argue that the gravitational path integral computation provides information about the typical magnitude of the overlaps for a collection of quantum states which only differ in finegrained structural details.The gravitational semiclassical description is insensitive to these fine grained details, but sensitive to the sizes.
The key insight, expanding on [10], is to rely on the fact that the Hamiltonian of the black hole is expected to be chaotic (see e.g.[16,17]), and that the thin shell that lives inside has a simple description in the local effective field theory basis of the black hole interior, i.e., it just creates a collection of particles.The behavior of the matrix elements of such simple operators O in chaotic theories is encapsulated in the Eigenstate Thermalization Hypothesis (ETH) [11,12] where the |E n are energy eigenstates of the black hole.
In this expression we chose an operator with no diagonal part because the shell operators of interest to us have will have a vanishing expectation value in energy eigenstates, to leading order.To see this, note that (18) can be regarded as an overlap between a one-shell state and a zero-shell state, a process which will will be exponentially suppressed in the mass of the shell.Equivalently, we can choose any operator and subtract the diagonal to create another operator.We also defined Ē ≡ (E n + E m )/2 and ω = E m −E n .The function g( Ē, ω) is smooth in the thermodynamic limit, and it encodes information about the microcanonical two-point function of the operator.The coefficients R nm are erratic complex numbers of O(1) magnitude.The ETH then asserts that the R nm entries can be viewed as independent random variables with zero mean and unit variance.This widely accepted hypothesis naturally provides an effective ensemble of operators which approximate the coarse-grained properties of our shell operators in the semiclassical saddlepoint approximation.In our case, this effective ensemble describes an ensemble of microstates of the black hole which have identical semiclassical descriptions.We now return to the moments of the inner products, Ψ m1 |Ψ m2 Ψ m2 |Ψ m3 ... Ψ mn |Ψ m1 .In this expression, the overline denotes a computation based on the gravity path integral.By writing every state in terms of the shell operators used to prepare them, and assuming ETH for the thin shell, we can interpret the overline in terms of an average of the form where In this way, we arrive at a simple interpretation of the non-factorization of the gravitational inner products for our microstates: the semiclassical path integral only computes averages over the ensemble of operators with the same semiclassical description.In other words, the operators in the ensembles all describe very similar states which differ in, say, Planck or string-scale details concerning the placement of the particles in the dust shell.The fine-grained phases of the states in the overlap Ψ m |Ψ m for m = m will depend erratically on the fine-grained choice of matrix elements of the operators O m and O m in the energy basis.Thus this overlap averages to zero semiclassically, Ψ m |Ψ m = 0, over the ensemble of states with the same macroscopic description.However the magnitude of the overlap is a self-averaging quantity because the phases cancel, and its typical value is captured by the wormhole contribution in (8).Equivalently it is captured by the exponentially small but still non-vanishing ETH correlations (19).

Eigenvalue density of the Gram matrix
To compute the rank of the Gram matrix G, we need to count the number of non-zero eigenvalues.We can compute this from the resolvent R ij of G ij : (20) The eigenvalue density of the Gram matrix is determined by the discontinuity across the real axis of the trace of the resolvent R(λ) = Ω i=1 R ii (λ).More precisely, the eigenalue density is On the right side of the resolvent equation (20) we have a sum of moments of the Gram matrix, or equivalently, of moments of the overlap between shell states.We can use the gravitational path integral to compute these overlaps writing R ij (λ) in terms of (G n ) ij where the overline implies that we have used the Euclidean saddlepoint approximation, interpreted as described above in terms of an ETH ensemble of states with the same macroscopic description.For these types of matrix Taylor expansions, as shown in [18] (also see [4]), one can write down a Schwinger-Dyson equation for the trace of the resolvent: where Ω is the dimension of the Gram matrix, and we used the result from above that As described in the main text, we constructed microstates of the black hole with fixed temperature, rather than fixed mass.Thus we should project the states, and hence the Gram matrix, into a microcanonical energy window.We can do this via an inverse Laplace transform [4] Z bh (nβ) = dE z(E) e −nβE . (24) For an energy window [E, E + ∆E], we define the functions As shown in [4], provided the wavefunctions of the states as expressed in the energy basis have statistically uncorrelated phases, the moments of the Gram matrix constructed above decompose into a sum of the moments of the Gram matrices associated to each microcanonical window.It follows from this that the resolvent of the projected Gram matrix will also satisfy ( 22), but with Z n inserted instead of Z bh (nβ).We can now perform To indicate that the computation of the resolvent was done using the semiclassical gravity saddlepoint we have included an overline above D(λ).

FIG. 2 .
FIG.2.Induced geometry at the time-reflection symmetric slice Σ.The horizon is at H = H + ∩ H − .The maximum surface σ = Σ ∩ W represents the position of the shell, at a radius R * ≥ rs inside the black hole.The geometry inside the shell caps of smoothly, and it is a portion of flat spacetime.

FIG. 3 .
FIG. 3. Euclidean continuation of the spacetime geometry of the microstates along Σ.The Euclidean section consists of an Euclidean black hole (right), and a region of Euclidean flat space (left), glued together along the trajectory of a thin shell.The shell starts at the asymptotic spatial infinity, bounces back at R * and returns to R = ∞.The total periodicity of the boundary equals 2πβ where β is the inverse temperature of the black hole.The euclidean times βm, β m ≤ β depend on the mass of the shell (see text).

FIG. 4 .
FIG.4.Euclidean wormhole contribution to the second moment of the overlap.The wormhole connects two asymptotic boundaries which prepare the overlaps.It consists of a portion of the Euclidean black hole on the left, glued to a portion of Euclidean flat space on the right along the two shells.

the sum and write down a quadratic equation for the trace of the microcanonical resolvent R(λ) 2 +
e S − Ω λ − e S R(λ) + Ω e S = 0 ,(26)where we have defined e S ≡ z(E) ∆E.Here S = A/4G N as discussed in the main text.Following (21) the density of eigenvalues is ) Ω − e S θ(Ω − e S ) .