Black hole evolution in a quantum-gravitational framework

Abstract

1. Introduction

The lack of gravity quantization in a renormalizable manner has long been an obstacle to progress that would otherwise have been possible. Various reviews of approaches to quantization can be found in Refs. [1–14]. Due to the series of recent developments in gravity quantization in which renormalizability of the physical states has been established [15–17], however, one is no longer shackled by the off-shell non-renormalizability. The Firewall argument [18,19] is a recent reminder that all is not well in our previous understanding of physics involving gravitational phenomena, and in particular black hole physics. (Reviews on black hole physics can be found in Refs. [20–22].) Arguably, the most significant contribution of the Firewall is that it has shaken several widely adopted views in the field, leading to various further developments. One of the views that the Firewall argument has put under close scrutiny is the conventional interpretation of the Equivalence Principle as implying a smooth and structureless horizon. The purpose of the present work is twofold: the first is to collect various ideas in the literature related to black hole information [23] (reviews can be found in Refs. [24–27]) and Firewall, and put them in a quantum-gravitational perspective; the second is to present a quantum-gravitational framework of black hole evolution, i.e. formation and evaporation, with the aim of tackling the black hole information (BHI) paradox, among other things. In particular, the ideas put forth in Refs. [28,29] on the relevance of the pre-Hawking radiation and vacuum decay to BHI are made precise.

The quantum gravitational setup to tackle BHI that we employ is along the lines of the scattering amplitude framework [30], and in that sense it is not new. However, we will pay special attention to the pre-Hawking modes and boundary dynamics, as well as the role of a bounce wormhole solution. In our analysis of black hole evolution, all of the types of system information are manifestly represented from the outset: the pieces of the information carried by various components appearing in the scattering amplitudes, such as the non-perturbative contributions, are explicitly (at least, conceptually) exhibited. The system information is classified into three categories: perturbative information associated with the Fock oscillators, information associated with the instanton-type non-perturbative contribution, and information that can be extracted through black hole perturbation analysis. All of these pieces of information are preserved, and unitary evolution is maintained throughout.

The focus of the quantitative analysis of the present work is the construction of the bounce solution, computation of its entropy, and its role in the information paradox. These are central to the resolution of the BHI paradox. (For the other two categories, we discuss their status and tasks that need to be carried out for a complete picture.) The non-perturbative contribution to the path integral is dominated by a solution that is nothing but a black hole analogue of the Coleman–De Luccia bounce solution [31,32]. Although the connection to the Coleman–De Luccia bounce solution is newly proposed in the present work (as far as we can see), the same name, “bounce,” has been used for “Lorentzian bounce solutions” in the gravitational context [33,34].

There are several critical ingredients that have led to the outcome of the present work. First is the recognition of the existence and role of the “pre-Hawking” radiation [28]. The pre-Hawking radiation, noted by Hawking himself [23], is an important part of the system components, and will be one of the main players in the entanglement of the system components. In addition to the information perturbatively stored by the Fock oscillators acting on the states, a substantial part of the information is stored in the entanglement between the components of the system, and to properly account for the entanglement, all of the system components must be counted. Second is the identification of the Hawking radiation: we propose that the particles generated by the vacuum decay be identified with the Hawking radiation. Particle generation by a time-dependent geometry is a well-known phenomenon [12]. In the present context it is the bounce solution that generates the particles that in turn are identified with the Hawking particles. Although being overly simple and thus having some undesirable features, this captures the essence of the mechanism of information release. Third is the relevance of the boundary degrees of freedom and their dynamics. (An earlier related discussion on the boundary dynamics in the context of BHI can be found in Ref. [35].) It has been competently demonstrated by a variant of Kaluza–Klein reduction that the boundary has non-trivial dynamics [36], and the Hilbert space of the bulk theory must be enlarged accordingly [37] (see Ref. [38] for an earlier related observation), which in turn is crucial for gravity quantization and surrounding matters, including BHI. These ingredients naturally become relevant as we proceed with the quantum-gravitational setup presented in the main body.

The boundary configuration plays a critical role in choosing the in and out vacua. To study the black hole forming process and its information perspective, the in-state is chosen to be a white hole and the out-state is built on a collapsing shell or shock-wave (not the other way around; see the comments in Sect. 3). As we will detail, the states of the boundary theory dictate the bulk boundary conditions. The non-perturbative contribution to the path integral is dominated by the bounce solution. It represents a shell that collapses from an initial boundary condition associated with the white hole and subsequently expands; it bridges the vacuum transition from the white hole state to the black hole state. Our toy model presented in the main body serves the purpose of illustrating the key ideas, with minimal technical complications.¹

Several remarks are in order for the bounce solution and its physics. One thing worth noting is that in the present description of the Hawking radiation it is not necessary to construct a bulk solution that explicitly realizes a shrinking and evaporating black hole. This part of the physics is accounted for by the bounce wormhole solution as a non-perturbative contribution to the path integral. (See the related comments in Sect. 3.3.) More specifically, it is the white whole state that accounts for the evaporated state: when the black hole evaporates, the mass is not lost in the case of a Dirichlet boundary condition, but converted into the energy of the white hole state. In other words, it is the novelty of vacuum decay physics that allows one to bypass explicit construction of a shrinking bulk solution. Another interesting feature of our analysis is that Wald entropy computed by using the bounce solution reveals a “shifted behavior” compared to the Page curve [39].² For instance, the Wald entropy takes a maximum value when the area of the bounce solution is at a maximum, whereas the Page curve starts there (namely, takes zero). The fact that there is such a shift is natural, given that the Wald entropy should count the entanglement from the pre-Hawking radiation as well. We expect that this feature will remain valid even for more realistic bounce solutions; the feature should be taken as a prediction of the present vantage point.

The rest of the paper is organized as follows. In Sect. 2 we start by surveying some of the salient issues regarding black hole evolution and information. We try to connect various ideas on BHI in the literature, especially those that lead to a non-smooth horizon. In Sect. 3 we first present the setup of the scattering of particles around a black hole. The non-perturbative decay process requires careful examination. For that, we consider a simple bounce solution to avoid technical intricacies. As it stands our solution is Lorentzian; Euclideanization is briefly discussed. Despite the fact that it has certain undesirable features, it captures at least some of the essential aspects of the entanglement through the Wald entropy. We discuss the issues surrounding the solution, and comment on a more realistic bounce solution. In Sect. 4 we ponder different types of black hole hair. The hair associated with the entanglement among all of the system components is crucial for the information problem. It was anticipated in Ref. [29] that vacuum decay will play an important role in BHI. We make this idea concrete. The Wald Noether charge measures the entanglement between the black hole degrees of freedom and the rest of the system [40,41]. (See also the earlier related work in Ref. [42].) The bounce solution will be the main contributor to the Wald entropy, and a “Page-like” curve is generically generated. We note that the Wald entropy becomes time dependent, despite the fact that it is a Noether charge, and explain why. The computed Wald entropy has an important difference compared to the Page curve: it is shifted. For a quantum-field-theoretic extension of the Wald entropy we consider its vacuum expectation value; the result will correspond to the quantum-corrected entropy. We also draw a connection with recent works in which the Page curve was produced. Section 5 has a summary and discussion of the possible implications of our results. We end with future directions. Appendix A contains some details regarding the bounce solution in Eq. (6) and its generalization.

2. Salient issue surrounding BHI

In this section we will provide a bird’s-eye view of some of the salient issues regarding black hole evolution and information. We relate them to some of the ideas on a non-smooth horizon and BHI in the literature. The vacuum decaying process requires a more extensive treatment and will be taken up in the next section.

Although Hawking’s original work employs a collapsing shell, a simpler setup for demonstrating evaporation of a black hole is through an eternal black hole. Let us remind ourselves, by taking an eternal Schwarzschild black hole, of the backbone of the BHI paradox in the semi-classical description. Afterwards we will turn to a time-dependent case. As is well known, there are several well-established coordinate systems, such as the Schwarzschild and Kruskal–Szekeres (Kruskal, for short) coordinates, adapted to observers in their states of the motion. Let us consider a free massless scalar field in the Schwarzschild and Kruskal coordinates, respectively. After semi-classical quantization of the scalar field, what leads to the Hawking radiation is the fact that the vacuua defined by the Schwarzschild observer and Kruskal observer are not equivalent [5,43]. Because of this, the Kruskal vacuum will appear to be radiating to the Schwarzschild observer. The radiation is thermal; the black hole will continue radiating until it disappears, leaving the thermal radiation behind. At face value the picture implies that an initial pure state has evolved into a mixed state, thus violating unitarity; the information seems lost.

There are several spots needing improvement in the account above. For instance, given that the evolution of a pure state to a mixed state characterizes the loss of information, is all of the system information lost? Where did the system information associated with the constituents of the infalling body (e.g. a collapsing shell) go? Even prior to these questions, what is the system information in the first place, since it must be more than the information carried by the entanglement among the constituents (although that part is central to the BHI paradox)? These questions suggest that a precise definition of the system information is desirable, and this is one of the things built in to the fully quantum-gravitational setup. As we will see, it is the information associated with the entanglement that is hardest to see preserved, whereas preservation of the rest of the information is not as hard. Another less obvious place for improvement is the manner in which the semi-classical analysis treats the boundary dynamics: it simply misses the dynamics entirely. A series of recent works [29,36,37,44] show that the boundary dynamics are critical for the information problem as well as gravity quantization. As we will see, the boundary dynamics³ play a crucial role in resolving the paradox.

Various works in the past, including Hawking’s original work [23], show that certain pieces of information escape at various stages of black hole evolution. In Ref. [23] it was noted that prior to the Hawking radiation, radiation in super-radiant modes comes out.⁴ (These types of pre-Hawking radiation will be important later in our picture of the proposed resolution of the paradox.) The Hawking radiation emerges only at a very late stage of the collapse [46]. To identify the various pieces of information involved in the black hole information problem, it is necessary to carry out the analysis at various levels: the Hawking-type analysis of the kinetic terms, quantum-field-theoretic analysis including the interactions, and analysis of non-perturbative contributions. In Sects. 3 and 4 we will systematically place these in the overarching framework of the scattering theory.

As we discuss in Sect. 3, the escape of some pieces of information can be easily recognized. This, however, is not the case for the Hawking radiation. Although it has long been expected that the Hawking radiation will be purified by some other components of the system, the detailed mechanism has not been clearly revealed. The puzzle culminates with the Firewall argument: when crossing the horizon of an old black hole, an infalling observer experiences trans-Planckian radiation.

Were it not for the Equivalence Principle, the Firewall would perhaps not be entirely surprising since the Schwarzschild vacuum would appear radiating to an infalling observer and the energy of the radiation should become more and more blue-shifted as the observer falls deeper in. Since the entanglement is expected to play an important role and constitutes a substantial part of the system information, one may investigate deeper into its involvement at the microscopic level. It is possible to understand this through a more systematic setup by employing a tripartite system. This was what was done in the Firewall argument. (As a matter of fact, there exist pre-Firewall indications of a non-smooth horizon in the literature; see, e.g., Ref. [47].) The backbone of the Firewall argument is that at the quantum level, it is impossible to have entanglement between the black hole interior degrees of freedom and those of the late Hawking radiation, which is necessary for a smooth horizon [48].

A more careful examination of the Firewall setup was given in Ref. [49]. What was shown there is that the near horizon can never be a vacuum due to the entanglement among all of the system components, the early and late Hawking radiation and the remaining black hole. The quantum effects lead to a non-smooth horizon: the quantum principle of entanglement is fundamentally incompatible with the Equivalence Principle (more precisely, the interpretation of the Equivalence Principle as implying a smooth horizon). The entanglement among all of the system components will be important in our discussion of BHI in Sect. 4.

Black hole information can be (and has been) tackled from different angles. For example, in the relatively recent works of Refs. [50–52] (that we became aware of sometime after finishing the present work), BHI was studied by employing the functional Schrödinger formalism, which we believe to be the most orthodox framework for studying some of the BHI- and Firewall-related issues. The authors analyzed the trace of the density matrix squared, and it was shown that once the quantum-field-theoretic interaction and correlation are taken into account, unlike in Hawking’s original works, the result was consistent with unitarity.

3. Quantum-gravitational framework

Setting up the full quantum-gravitational framework itself does not require intensive effort: it can be done within the realm of the standard quantum field theory. The framework is essentially scattering amplitude analysis including non-perturbative contributions. As we will see, the non-perturbative contributions are crucial for tackling the black hole information paradox. Since the perturbative part of the calculation can be handled in the recently proposed Feynman diagrammatic manner [17], we will focus mostly on the non-perturbative contributions.

The boundary dynamics are important for both quantization and BHI [36,44,53]. In Sect. 3.1 we qualitatively review the recent result obtained in Ref. [36] in which a reduction called the dimensional reduction to a hypersurface of foliation was carried out as a part of gauge-fixing.⁵ This sets the stage for Sect. 3.2, where we outline the scattering amplitude analysis. We write down the scattering amplitude in a schematic notation. A given scattering amplitude factorizes into perturbative and non-perturbative parts. Although technically complicated, the perturbative part can, in principle, be evaluated. As we will see, the non-perturbative part is reminiscent of the vacuum decay through the Coleman–De Luccia bounce solution [31,32]. In Sect. 3.3 we explicitly construct a bounce solution that is based on a shock-wave solution of an Einstein–Hilbert action with an appropriate stress–energy tensor. (Some details can be found in the appendix.) We discuss the subtleties in regarding our solution as a Coleman–De Luccia bounce solution. We also discuss the relation of our solution to the “Lorentzian bounce solutions” of Refs. [33,34,55–58]. The bounce solution will be of critical subsequent use.

3.1. Roles of the reduced action as the boundary theory

Before getting to the quantum-gravitational scattering setup, we review some of the recent developments that we will use in the following subsections and Sect. 4. The main thing to be reviewed is the relevance of the boundary action obtained while quantizing the bulk theory [17]. In Ref. [17] (and the references therein) it was noticed that the boundary theory is naturally obtained in quantizing the bulk theory as part of a gauge-fixing procedure. One can show that as a result of the gauge-fixing, the physical degrees of freedom of the bulk theory are realized on a hypersurface at the boundary.

As a matter of fact, the boundary degrees of freedom are part of the bulk degrees of freedom, and the boundary theory is part of the bulk theory (in that it describes the on-shell fluctuations of the bulk geometry; see below). This may sound incompatible with the standard anti-de Sitter / conformal field theory (AdS/CFT) in which the CFT is a priori independent of the bulk theory. This is not so. Instead, what was noticed in Ref. [17] (and earlier works) should be taken as a refined view of how AdS/CFT works: it must be the way in which the holographic dualities are played out. In the case of the best-established AdS|$_5$|/CFT|$_4$| case, it has been shown that the boundary theory, CFT|$_4$|⁠, can be obtained as the theory of the bulk moduli field [44]. Once one considers a given background of the bulk theory, the reduced action describes the fluctuation modes of the physical degrees of freedom. In other words, the boundary theory comes to describe the fluctuations—which is part of the bulk theory physics—around the on-shell background geometry. This is why the boundary description is dual to the bulk description: the former is the latter for the on-shell physics. The given boundary configuration determines the boundary conditions of the bulk theory.⁶

The boundary theory plays several significant roles in the discussions that follow: in Sect. 3.2, it is involved in choosing the in- and out-states in the scattering setup. In particular, a given configuration of the boundary theory will serve as the boundary condition of the bulk bounce-type solution that dominates the non-perturbative contribution to the amplitude. In Sect. 3.3 the boundary terms and conditions will be important for vacuum transition probability. In Sect. 4, where we ponder various pieces of information, the boundary theory will be carefully taken into account when scrutinizing the information pertinent to the entanglement between the bulk and boundary components.

3.2. Scattering setup

Let us first examine Eq. (2) for the different types of information contained. There are three different types.⁷ First, the “perturbative information” associated with the quantum numbers |$\{\alpha\}$| and |$\{\beta\}$| is preserved (say, through various Kronecker deltas or delta functions) in the usual way of calculating the Green’s function and S-matrix. Secondly, there is the information coming from quantum corrections. As we will review in Sect. 4, this is the information stored by the quantum boundary modes. These are the modes responsible for trans-Planckian energy near the horizon [59,60]. The third type of information is that associated with the entanglement between the constituents of the black hole and the rest of the system. The preservation of this portion of the information is most subtle to see.

Let us ponder the possible choices for |$|in;\{\alpha\}\rangle$| and |$|out;\{\beta\}\rangle$|⁠. Since we are interested in quantum-gravitational black hole formation, it is obvious what state to take as final: a certain black hole state. “Certain” because its hair will depend on the initial state of the boundary. A typical out-state will be the one obtained by applying Fock creation operators on the out-vacuum. For the in-state, one has a large amount of freedom even for infalling matter with spherical symmetry. For simplicity, one may consider choosing a spherical shock-wave infalling from the asymptotic region.

At this point, it is intriguing to note the relevance of the vacuum decay analysis of Refs. [31,61], where it was shown that the vacuum decay is mediated by a non-perturbative solution, the so-called “bounce.” (See, e.g., Ref. [32] for a review.) The vacuum decay amplitude is the probability amplitude for an initial false vacuum to decay into the true vacuum. It is tempting to view the black hole formation process from an initial state as analogous, with the black hole state being the true vacuum and the white hole state the false vacuum. One conceptual obstacle to this is the fact that the black hole formation process is not a priori a potential-tunneling process. There are, however, some indications that support the analogy. For instance, it was shown in Ref. [31] that the vacuum decay of a non-gravitational scalar theory can be embedded in a gravitational setup. More importantly, the path integral evaluation of the amplitude above seems to naturally suggest the relevance of a gravitational bounce-type solution. Based on these, we propose that the vacuum decay physics under present consideration be viewed as a generalized vacuum transition in the black hole context. (More on this in Sects. 3.3 and 4.)

Although there should be diverse initial states that one can take, one choice stands out for its technical simplicity. (A more realistic solution would be one based on an Oppenheimer–Snyder solution; see, e.g., Refs.[57,58].) Recall that a bounce solution has a bouncing behavior and is time-reversal symmetric before and after the bounce. A boundary configuration (and thus the corresponding bulk theory boundary condition) that leads to a simple bounce solution is the one dictated by a white hole final state after the bounce. Construction of a bounce solution that corresponds to more realistic collapsing matter will be much more involved. Fortunately, the essential physics can be understood by taking our bounce solution, to which we now turn.

3.3. A bounce solution

Fig. 1

Gravitational collapse of a shock-wave.

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Fig. 2

A bounce solution representing a shell that collapses from an initial boundary condition associated with the white hole and subsequently expands. The dotted line represents the event horizon; the segment a–b–c lies outside the shell. The entanglement entropy increases over the a–b segment whereas it decreases over the b–c segment.

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A word of clarification is in order. The |$|in;{\alpha}\rangle$| state represents a false vacuum state, and this is what was meant by the statement in the introduction that the bounce solution represents a shell that collapses from an initial boundary condition associated with the white hole. This state is chosen to be the time reversal of the usual white hole state represented by the upper half of the bounce solution in Fig. 2. (This structure is of course part of how Coleman’s bounce solution [61] works.)

Issues surrounding the bounce solution Before analyzing the Wald entropy of the bounce solution in Eq. (5) in the next section, we examine issues surrounding the bounce solution. Let us start with conceptual ones. To describe the solution we are using the term “instanton-type” contribution in the sense that the bounce solution makes a non-perturbative contribution. However, its Euclidean origin is not entirely clear. One may consider a Euclidean counterpart by analytically continuing the time coordinate |$t$|⁠, with resulting |$u,v$|⁠. For its full justification, however, it will be necessary to consider a matter part, such as a scalar field, as we further comment on below. Another conceptual aspect is that since we are considering the bounce as a non-perturbative contribution to the path integral, it is a classical solution. One can of course try to obtain the quantum-corrected bounce solution by solving the one-particle irreducible (1PI) action once the action is obtained (in a truncated form). Presumably this approach may well qualitatively confirm the results obtained along the lines of Refs. [33,34], where solutions that reflect quantum effects were constructed.

One technical issue is the undesirable feature of our bounce solution: containing |$|t|$|⁠, the solution is not analytic at |$t=0$|⁠. Although it should be possible, at least as an ad hoc measure, to realize the bounce solution as a limit of a sequence of, e.g., sigmoid functions, and avoid the feature, one may also try to avoid such a singularity by constructing a less artificial and more realistic solution. As a matter of fact, this appears to be what has been achieved in Refs. [57,58] by constructing a bounce solution based on an Oppenheimer–Snyder solution.⁸ As discussed therein, Planckian-scale physics is expected near the transition point.

The analysis above is based on the Dirichlet boundary condition, and as previously mentioned, one must be careful with the boundary conditions. The importance of boundary conditions has been noted in several different contexts. For instance, it was noted in Ref. [64] that there are potentially infinitely many different conditions with associated boundary terms, and quasi-local charges take different values depending on them. As mentioned in Sect. 1 and footnote ⁶, for quantization the Hilbert space must be expanded to include all of the different sectors of possible boundary conditions. One of the implications of Refs. [29,36,37] is that the well-known Dirichlet boundary condition is very special in that it should be associated with the “vacuum” of the three-dimensional dynamics. Having a common bulk solution, all of the different boundary conditions must be associated with different aspects of the bulk fluctuation reflected on the boundary [36]. This status of the matter suggests that in calculating a physical quantity, such as the decay time, one must take an “ensemble” average over the boundary conditions. Also, in light of the analysis in Ref. [36] in which it was shown that in general Noether charges are not conserved for non-Dirichlet boundary conditions, the mass will not be conserved for a generic boundary condition. Since the mass value will depend on the boundary terms and conditions, it will not, in general, turn out to be the same as the mass parameter |$m$| for a non-Dirichlet-type condition. Instead it is natural to expect the value to be smaller, reflecting the mass loss by Hawking radiation. It will be of great interest to see whether the “ensemble” average may lead to the |$m^2$| or |$m^3$| scaling of the decay time.

4. Black hole hair

With the discussions in Sects. 2 and 3, we are ready to take an in-depth look at the system information and its preservation. Examination of the information should be carried out at various stages of the quantum-level analysis. In particular, the information stored in the boundary configuration becomes relevant at the early stage of quantization when determining the physical states, as well as later when choosing the in- and out-states. The next stage is the actual evaluation of the scattering amplitudes. The perturbative piece of information, which is associated with the Fock space creation operators, can be extracted in the standard manner (though this is highly complicated technically).

As for the non-perturbative information, there are two types. The first non-perturbative information is associated with the instanton-like solution called the “bounce.” The name was originally introduced in a non-gravitational quantum field theory, in fact, in the quantum-mechanical context [61]. In the gravitational context, the same name was given to a class of solutions that have a “bouncing” feature [33,34]. It appears that the connection between the two types of bounce solutions was not recognized in the literature. We propose that the gravitational bounce solution be viewed as having the same origin as the original bounce solution: it is associated with the vacuum transition. The second type of non-perturbative information can be extracted by analyzing the 1PI effective action. Except for the fact that the object to be analyzed is the 1PI action, the analysis is in the same spirit as in quasi-normal mode calculation: the analysis employs the black hole perturbation method. What information can be extracted through such analysis? For this, it is useful to start with a classical-level analysis. The presence of non-trivial matter field configurations represents part of the system information. In addition, the standard classical black hole perturbation reveals that the radiation takes away all of the multipoles except the first several associated with mass, charge, and angular momentum. With these approaches elevated to the quantum level, one should discover that the quantum boundary modes carry a significant amount of the system information (see below). This should be the quantum-gravitational bleaching anticipated in Ref. [28].

Below, we scrutinize each of these two sectors. In particular, for the information mediated by the bounce solution, we examine the entanglement between the black hole degrees of freedom and the rest of the system. The entanglement information should be measured by the Wald entropy charge. It is important to note that in considering the bounce solution, which is time dependent, one can nevertheless consider the time-like Killing vector after the shell crosses the expanding event horizon. This is because, according to Birkhoff’s theorem, the geometry outside the expanding spherical region should still be the static Schwarzschild geometry. A shifted Page-like curve is generically produced. Elevating the Wald entropy charge to the quantum level is straightforward: it is the vacuum expectation value of the charge, which can be calculated by applying Wald’s method to the quantum-corrected action.

In Sect. 4.1, the information mediated by the bounce solution is analyzed. In Sect. 4.2, recent results in black hole perturbation are reviewed with a focus on the information carried by the boundary quantum modes. In Sect. 4.3, we combine the results and present an overall account of black hole formation and its information release pattern.

4.1. Information mediated by a bounce

We have discussed that the dominant contribution to the non-perturbative part of the scattering amplitudes will come from a bounce solution. For a convenient visual illustration with minimum technical complications, one can use the null-shell-collapse-based bounce solution explicitly constructed in the previous section. Ideally, one should consider an Einstein–Hilbert action coupled with matter, say, a scalar field, instead of the stress–energy tensor engineered by the choice of the metric. Then the in- and out-states would belong to different non-perturbative sectors of the theory. What we propose is that the transition between two states belonging to different sectors may be analyzed along the lines of Ref. [61]. In fact, in the case of the field theory analysis in Ref. [31], it was shown that the vacuum decay physics of a non-gravitational scalar theory, which is a transition from a false vacuum of the potential to the true vacuum, can be embedded in a gravitational setup. In any event, it should be possible to construct a bounce solution in an Einstein–scalar system. (For instance, a collapsing-shell solution of an Einstein–scalar system was numerically constructed in Ref. [65].) It is just that we employ the bounce solution constructed in the previous section to avoid inessential technical complexities.

Recall that, as we have reviewed in Sect. 3.1, the boundary condition of the bulk theory is tied with the boundary theory state and that boundary theory describes the physical degrees of freedom of the bulk theory in the given background. As we saw in Sect. 3, a technically simple choice of the boundary state is associated with the bounce solution that has the black hole state as the true vacuum and the white hole state as the false vacuum. In the path integral evaluation of the amplitude, one should consider expanding the action around the bounce solution. The leading-order action, i.e. the action evaluated on the bounce solution, can be pulled out of the path integral; it is the leading non-perturbative contribution.

Another role of the bounce solution is that, being a time-dependent geometry, it generates particles [66,67]. This is not entirely surprising since such particle generation is well known in cosmology [5,68]. Moreover, it is those particles that were identified as (what is now known as) Hawking radiation in Hawking’s original work. That interpretation should remain valid in the present non-perturbative context: the particles generated by the vacuum decay should be the Hawking radition.⁹

Toward the end of this section, we will carry out entropy calculation and observe a shifted Page-like curve originating. Let us try to develop some intuition before getting to the quantitative analysis. When considering the entanglement among the system components, it is crucial to take into account the modes represented by the Fock oscillators since they too will be entangled with the Hawking radiation. The entanglement between the black hole degrees of freedom and the rest of the system, including not only the Hawking modes but also the hard (and soft) modes, must be considered. This is because everything in the initial infall must have been entangled, and thus should remain as such. (See Refs. [69–71] for recent related works.) As stated previously, this must be the reason for the fact that the resulting curve is shifted as compared to the Page curve.

Lastly, being too simple, our bounce solution misses some aspects of the physics associated with black hole evaporation. A more realistic solution would also properly take the quantum-gravitational corrections into account. (The fact that one considers the instanton-type contribution through the bounce solution means that non-perturbative corrections are being taken into account. In addition to this, one must consider |$\hbar$| corrections to the bounce solution itself.) To capture such physics it will be crucial to start with a more realistic solution, which we describe in the next subsection.

4.2. Nonperturbative hair revealed by BH perturbation

Another line of the system information can be extracted through standard black hole perturbation theory. At the classical level, black hole perturbation theory was developed long ago (see, e.g., Refs. [20,22] for reviews.) The classical quasi-normal mode (QNM) analysis reveals that a black hole radiates away most of its multipole characteristics, except the ones associated with mass, charge, and angular momentum. It should be possible to extend black hole perturbation theory to the quantum level. Repetition of the QNM analysis should again show that the higher-order multipoles will be radiated away.

By changing the methodology somewhat, one can explore different aspects of the system information. The method is quite suited when it is the quantum corrections that induce the perturbations. Applied to the present case, one may try to obtain the quantum-corrected bounce solution by solving the quantum-corrected action. However, although the bounce solution serves well the purpose of analyzing the first type of non-perturbative contribution, there are simpler solutions for illustrating the second type of non-perturbative contribution presently being considered: they are the familiar static or stationary solutions such as a Schwarzschild or Kerr solution. Thus we leave the bounce solution in the remainder of the present subsection.

The motivation for Refs. [59,60] was the Firewall. The tripartite system that split out of the original black hole makes it clear that one cannot have the required entanglement for a smooth horizon between the late Hawking radiation and remaining black hole degrees of freedom [49]. An infalling observer will observe highly blue-shifted radiation near the horizon if the area is not vacuum. The results in Refs. [59,60] seem to be a computational confirmation of the presence of a Firewall-type structure at the horizon.

4.3. Information-release pattern

In the previous sections we have collected several ingredients. In Sect. 3 we defined system information in order to study the evolution (namely, the formation and evaporation) of a black hole in a quantum-gravitational scattering setup. We also explicitly constructed a bounce solution that determines the non-perturbative contribution in the path integral. In Sects. 4.1 and 4.2 we proposed that the particle generation under the vacuum decay be identified as the Hawking radiation, and part of the system information is captured by various black hole perturbation analyses in the spirit of the QNM modes. One gets the following picture of black hole evolution as an upshot of putting these ingredients together. Initially, an incoming shell with perturbative excitations by the Fock creation operators starts collapsing, and at the same time the perturbative excitations—whose information is obviously preserved—are scattered away. While the collapsing body forms a black hole, most of the multipole moments, except the ones associated with the mass, charge, and angular momentum, are radiated away. Therefore, much of the system information heads for the asymptotic region. Once the black hole is formed, it starts to decay through the instanton-type non-perturbative channel, the bounce. In this subsection we examine, as the last piece of the complete picture, the information associated with the entanglement between the system components.

Although the information preservation seems obvious in the present picture, one should understand why the semi-classical analysis led to apparent information loss. It is essentially because in the semi-classical analysis only the black hole and Hawking radiation (early and late) were considered. As we have seen, when considering the entanglement among the system components it is important to consider all of the components. In other words, for a proper understanding of the purity and unitary evolution of the system, the pre-Hawking radiation and boundary dynamics must be taken into account as well [28,29]. There is an indication of this from another angle in the entropy analysis below.

Thus, |$r_*$| and |$r$| increase as |$t$| increases.

The quantum extension of the Wald entropy formula seems straightforward: instead of the classical action, one should use the quantum-corrected action.¹² Since the quantum-corrected solution is expected to preserve the bouncing feature of the classical solution, so should the quantum-corrected Page-like curve.

One significant implication of the analysis above is the relevance of the pre-Hawking radiation [28], which is governed by the boundary theory. The analysis above shows that black hole formation is a process in which the to-be-black hole degrees of freedom and the rest of the system are redistributed¹³ while remaining entangled and thus keeping the initial purity. As the shell collapses the event horizon appears and its area increases: this means that the black hole degrees of freedom become more and more entangled with the rest of the system while the whole system remains pure. In the bouncing-out phase of the shell, as more and more Hawking radiation (namely, the particles created by the decay) join the bulk, the bulk degrees of freedom almost constitute the overall state, which is pure, with less entanglement with the black hole degrees of freedom. This is reflected in the decrease of the area of the horizon.¹⁴

Although the present picture of black hole formation and evaporation shares qualitative features with Page’s account, it refines it in that the entropy starts to increase once the event horizon appears and to decrease once it reaches its full size. If we disregard the entanglement between the pre-Hawking modes and black hole mode, the entropy should increase for a while even after the black hole formation. However, if we take the pre-Hawking modes into account, there are already substantial degrees of freedom out, so the Hawking radiation makes the entanglement decrease.

5. Conclusion

In this work we have systematized black hole evolution in a quantum-gravitational scattering setup in which all of the different types of information are kept track of throughout. The results obtained can be summarized as follows. First, the qualitative picture (presented in the first paragraph of Sect. 4.3) of the system evolution. Second, the construction of the explicit bounce solution and its role in BHI. Third, the computation in regards to what the Wald entropy measures and its quantum extension. In the present quantum-gravitational scattering approach, it is evident that the information is preserved. The entanglement between the black hole degrees of freedom and the rest of the system can be measured by the Wald entropy charge; a shifted Page-like curve has been produced.

Compared with the semi-classical analysis, the present study has several key ingredients that have led to the information preservation. The first ingredient is the role of the boundary degrees of freedom. In recent works, it has been anticipated that consideration of non-Dirichlet boundary conditions will be crucial for resolving BHI. It was proposed that the Hilbert space should be enlarged to include the sectors of different boundary conditions. The in- and out-states that we have considered belong to the different sectors, and this naturally led to the consideration of a bounce solution. The second ingredient is in conjunction with the bounce solution that we have come up with: the generalization of the vacuum transition. For this, the general configuration space should be considered. Also crucial was the identification of the particles generated from the vacuum transition with the Hawking radiation. On a more technical side, employing the simple collapsing-shell-based bounce solution has facilitated the analysis. We believe that the bounce solution that we have constructed, though not entirely realistic, captures the essence of the problem.¹⁵ As for a more realistic bounce solution, we take Refs. [31,65] as an indication that the stress–energy tensor found in the present work can be realized in the context of an Einstein–scalar system.

One of the lessons learned is that the non-perturbative quantum effects (the instanton effect in the present case) are important regardless of large curvature. The fact that the quantum effects are important regardless of large curvature was also well demonstrated in recent works where a trans-Planckian energy results near the horizon of the time-dependent black hole [59,60].

5.1. Future directions

One intriguing aspect of the Noether current is the fact that certain components of the current are radially conserved [83]. Presumably there exists a flow of entropy whose flux is conserved [41,84,85]. It will be interesting to understand the potential implications of this for the Ryu–Takayanagi result [86].

As we have noticed, there are certain similarities between the Hawking modes and QNMs. Two things hamper a direction identification of these two notions. First, the Hawking radiation is mainly a q-number description whereas QNMs are in the context of a c-number description. Second, the QNMs do not constitute a complete set since they satisfy special boundary conditions [87] whereas the Hawking modes provide a basis. In spite of these differences, the quasi-normal modes should somehow be related to the Hawking modes [88], and making the relation more precise will be of some interest.

Another direction concerns the construction of a more realistic bounce solution. The bounce solution that we have considered is time-reverse symmetric. There may be other types of non-perturbative solutions that are not. Perhaps not unrelated, it will be interesting to construct a quantum-corrected bounce and see whether or not the solution will remain time-reverse symmetric. As discussed in Sect. 3.3, the transition probability has room for significant improvement. The task is tied in with enumeration of all possible boundary conditions, which is an interesting problem on its own.

Our final thoughts are on the following rather radical possibility. We have seen that the non-perturbative part of the scattering amplitude is dominated by the bounce solution. The bounce solution is obviously a time-dependent solution. Although we have not demonstrated it for a bounce-type solution, for a geometry obtained by perturbing a stationary geometry, such as a perturbed Kerr, we have recently demonstrated (as reviewed above) that generic perturbations, which should thus be time dependent, lead to a trans-Planckian energy. Perhaps a Firewall may be a way for Nature to avoid the curvature singularity. In other words, Nature may avoid the curvature singularity by quantum effects that produce a Firewall so that the singularity cannot be accessed.

Funding

Open Access funding: SCOAP|$^3$|⁠.

Appendix A. Some details of the bounce solution

Then, the |$(u,r)$| or |$(v,r)$| sector of |$G_{\mu\nu}$| will have to be matched with the corresponding block of the stress tensor constructed out of, say, a scalar system. Construction of a realistic solution will be further pursued elsewhere. In such a case, one may have to pay close attention to the boundary terms (see Ref. [9] for a general review).

Footnotes

References