The preëminent view that evaporating black holes should simply be smaller black holes has been challenged by the firewall paradox. In particular, this paradox suggests that something different occurs once a black hole has evaporated to one-half its original surface area. Here we derive variations of the firewall paradox by tracking the thermodynamic entropy within a black hole across its entire lifetime. Our approach sweeps away many unnecessary assumptions, allowing us to demonstrate a paradox exists even after its initial onset (when conventional assumptions render earlier analyses invalid). Our results suggest that not only is the formation of a firewall the most natural resolution, but provides a mechanism for it. Finally, although firewalls cannot have evolved for modest-sized black holes, within the age of the universe, we speculate on the implications if they were ever unambiguously observed.
pacs:04.70.Dy, 03.65.Xp, 03.67.a, 03.70.+k
The fundamental physics of black holes has been an enduring mystery Hawking76 (). Great progress has been recently made with the discovery of the firewall paradox for black holes, which suggests the existence of a manifestly strong phenomenon, the firewall AMPS () or energetic curtainBraunstein13 () as it was originally dubbed in 2009. Loosely, the paradox constructs a contradiction between the correspondence relating real to classical black holes, their thermodynamic behavior and quantum unitarity. However, finding the minimal assumptions has remained an open question.
Here, we provide a streamlined firewall paradox with many unnecessary assumptions removed. Notably, no measurement and consequently no decoding of the Hawking radiation is needed (making its complexity Hayden () irrelevant), nor do we rely on any specific radiation process like pair creation or tunneling. This last point deserves especial comment: First, if firewalls were real, then ‘nice time slices’ through a black hole’s spacetime Lowe95 () and all mechanisms associated with them could no longer be trusted. Second, black hole evaporation is strongly believed to be non-local (e.g., via the holographic principleHooft (); Susskind95 ()). However, pair creation in particular comes from local quantum field theory Hawking76 (). These considerations render earlier derivations of the firewall paradox AMPS () problematic. Next, we derive a new paradox which independently explores the role of non-local physics across the horizon. We find that there is an extra ingredient left out of conventional holographic approaches Maldacena (); Hooft (); Susskind95 (). Combined, these two paradoxes provide insight into black hole physics across their entire lifetime. Evidence supporting our assumptions and variations to them are given in the discussion.
Before proceeding, let us review an important tool: The quantum mutual information, , provides a measure of correlations between a pair of systems and . Here the von Neumann entropy, denoted , gives the thermodynamic entropy for an isolated systemvN (), . As with some earlier works studying black hole evaporation Mathur09 (); AMPS () we shall rely on the property of strong subadditivity Wilde (), which for any extra system , states that .
We will leverage this tool to identify contributions to thermodynamic entropy in even very large quantum systems. Suppose and are remotely separated and no longer interact. If the correlations between these systems correspond to maximally entangled subspaces of and , then they make a distinct contribution fn3 () of to the thermodynamic entropy in (and in ); an entropy that is observable by even highly delocalized detectors. This is in contrast to local correlations due, for example, to entropy of entanglement in the vacuum state across a boundary Eisert10 (). In that case, correlations are localized to a narrow layer at the boundary and unless one’s detectors are localized on scales comparable to the cutoff (presumably Planckian) such entropy is unobservable.
Non-exotic atmosphere: To reconcile gravity with quantum mechanics, it is generally assumed that there exists a correspondence between the physical characteristics of a real (i.e., quantum mechanical) black hole and its theoretical classical counterpart. The tightest correspondence Hawking76 () assumes that black holes evaporate into vacuum (as seen by an infalling observer). Here, our two theorems shall make much weaker assumptions than that the quantum field into which the black hole evaporates is in (or anywhere near to) a specific quantum state.
In part, we achieve this by focusing on the gross thermodynamic properties of the neighborhood, , external to and surrounding a black hole that reaches out far enough to encompass any process by which radiation is produced. Now, recall ’t Hooft’s entropic bound Hooft (); Hsu08 (), which shows that if one excludes configurations of ordinary matter that will inevitably undergo gravitational collapse, one finds , where is the surface area of the region containing that matter (in Planck units) and is the thermodynamic entropy of the matter (with Boltzmann’s constant set to unity).
Suppose that the external neighborhood, at some specific epoch, has a surface area times that of the black hole’s horizon, so , where denotes the black hole’s Bekenstein Hawking entropy. Then to satisfy ’t Hooft’s bound, the thermodynamic entropy of the external neighborhood, , must be bounded by
For large black holes, the prefactor in square brackets is much much less than unity even for extremely large neighborhoods, e.g., . If this bound fails, the external neighborhood, , must consist of some exotic matter, such as an atmosphere of microscopic black holes.
Theorem 1: A contradiction exists between: 1.a) completely unitarily evaporating black holes, 1.b) a freely falling observer notices nothing special until they pass well within a large black hole’s horizon, and 1.c) the black hole interior Hilbert space dimensionality may be well approximated as the exponential of the Bekenstein Hawking entropy.
Proof: Assumption 1.a has two ingredients (i and ii): That black hole evaporation is unitary and that it is complete leaving behind no remnant. We start by unraveling the implications of these two ingredients separately:
Unitary evaporation 1.a(i): Consider the unitary generation of radiation from a black hole by an arbitrary process, Fig. 1. We associate this process with some specific black hole and presume that to an excellent approximation radiation is not produced further out beyond .
Applying strong subadditivity to Fig. 1 trivially yields . Further, as entropy is invariant under unitary transformations and we obtain
This invariance has allowed us eliminate and from Eq. (2), thus allowing us to work with quantities and which may be associated with a still large black hole.
Complete evaporation 1.a(ii): Consider a black hole created by collapsing matter initially in a pure quantum state (the Appendices consider more general scenarios). After the black hole has completely evaporated away the net radiation, , should also be in a pure quantum state to preserve unitarity. Thus one might expect that correlations would exist between the early and late epoch radiation, and , respectively.
In fact, the study of random unitary operations allows us to say much more: Since the Hilbert space dimensionalities involved are so huge Levy’s lemma guarantees a generic behavior for entropy in all but a set of measure zero of evaporation mechanisms Braunstein13 (); fnPir (). In particular Page93 (); Braunstein13 (), the entropy of the radiation grows monotonically (at almost exactly the maximal rate of one bit’s worth of entropy per qubit of radiation emitted bit ()) until the Page time (when the black hole’s area has halved). From the Page time (PT) onward the overall entropy in the radiation decreases at the same rate, reaching zero when evaporation is complete Page93 (); Braunstein13 (). Thus, .
Now recall that the Bekenstein Hawking entropy quantifies the thermal entropy in a black hole from the first law of black hole thermodynamics. This presumes, however, that the radiation lacks correlations. Thus by ignoring correlations in the above results we may identify Manas () the Bekenstein Hawking entropy of the initial black hole as , where . Similarly, after evaporating radiation , the remaining black hole has Bekenstein Hawking entropy . Labeling the pre- and post-PT as and , respectively, then gives
For black holes formed by matter in a pure quantum state, the (global) state of may also be treated as pure implying . Because it will be sufficient to consider only really huge violations of assumption 1.b, we can decompose it into external and internal constraints (i and ii).
External free-fall equanimity 1.b(i): An exceedingly weak assumption is that a freely-falling observer notices no exotic matter at least down to the horizon. Thus, from Eq. (1), the radiation-correlated contribution to the neighborhood’s entropy must be negligible. Combining this with Eq. (2) yields
Finite interior Hilbert space 1.c: In order to ensure that the black hole interior (within the stretched horizon) can contain no more physical entropy than can eventually evaporate away, it is assumed that the interior has a Hilbert space dimensionality that is well approximated by the exponential of the Bekenstein Hawking entropy. This implies that during a black hole’s evaporation and hence from Eqs. (3) and (4) we have
So far there is no contradiction. However, Eq. (5) tells us that the black hole interior initially accumulates thermodynamic entropy at the rate of (at least) one bit per qubit radiated. From the PT, the shrinking interior (everything within the stretched horizon) is filled with half of a maximally entangled state, , corresponding to an infinite-temperature thermal state.
Internal free-fall equanimity 1.b(ii): A contradiction ensues if we assume that an infalling observer notices nothing (or at worst a low entropy state) well within the horizon. Indeed, after the PT there is nowhere left within the black hole (within the stretched horizon) where an observer can exist without intimate contact with the infinite-temperature interior.
This result follows straightforwardly and differs from earlier arguments AMPS () which invoke a likely impossible capacity for decoding Hawking radiation Hayden (). In the Appendices we extend our analysis to explicitly account for the negligible thermodynamic entropy in and to exclude the physics of Planck-scale black holes.
The AdS/CFT correspondence crisis: The strongest contender for a fully unitary theory of black hole evaporation involves the AdS/CFT correspondence Maldacena (), which formalizes the holographic principle Hooft (); Susskind95 (). Unfortunately, this theory gives no hint of a firewall nor that anything unusual might happen within the horizon from the PT onward. The conventional hope is that this discrepancy will be resolved once the “dictionary” relating the CFT and black hole interior states is better understood.
Despite this hope, this discrepancy has precipitated such a crisis for the AdS/CFT and holographic approaches that their creators have resorted to a radical Maldacena13 () (many in the community call it an absurd Susskind14 ()) solution. In particular, it is proposed that all maximal entanglement (denoted EPR) is associated with Einstein Rosen (ER) bridges (a kind of non-traversable wormhole) Maldacena13 (); it is claimed Maldacena13 () that this ER=EPR proposal provides a way to explain the disturbance which occurs in the original firewall paradox argumentAMPS () when the Hawking radiation is measured without implying the existence of a firewall.
As a putative counterexample to the firewall paradox, a scenario closely mimicing the manifestly smooth eternal AdS black hole Maldacena () was considered Maldacena13 (), consisting of a pair of ‘maximally entangled’ black holes connected by an ER bridge. Provided these black holes evaporate, it is straightforward to extend Theorem 1 to this scenario: Initially, there is no firewall, however, by the PT the paradox is reinstated. No measurements are needed and with no disturbance, ER=EPR is left without a role. (In fact, the ‘entanglement’ in this example is local vacuum entanglement across nearby horizons and vanishes in ; see the Appendices).
Notwithstanding this, Ref. Susskind14, may be interpreted as claiming that with the advent of the firewall argument there is at least something important missing in the conventional AdS/CFT description of black holes. Indeed, an enduring frustration with the AdS/CFT correspondence has been that it gives us no inkling of why the ‘nice time slice’ argument supporting local physics (even prior the PT) is apparently wrong. As such, it seems wise to guard against blind acceptance of intuitions about non-local physics coming from holographic approaches Maldacena (); Hooft (); Susskind95 (). It is therefore worthwhile to reëvaluate the role of non-locality during black hole evaporation. We do this here with Theorem 2. We shall see that any potential clash between unitarity and locality actually requires a third ingredient left out from holographic considerations.
Theorem 2: A contradiction exists between: 2.a) completely unitarily evaporating black holes, 2.b) large black holes are described by local physics, and 2.c) externally, a large black hole should resemble its classical theoretical counterpart (aside from its slow evaporation).
Proof: Note that assumption 2.a is identical to assumption 1.a of theorem 1.
Local physics 2.b: In addition to assuming the complete unitary evaporation of a black hole (2.a), we shall suppose that whatever process generates the radiation it is constrained to be local for large black holes. In particular, we shall focus on the fact that local physics forbids communication across light cones Strominger (), so that there can be no communication from within a large black hole’s event horizon to the exterior.
In order to make use of this non-communication property we recall the no-communication decomposition theorem Werner () (see Fig. 2) which tells us that any unitary process which does not allow communication from a set of inputs to a set of outputs may be decomposed into a pair of unitary subprocesses and with at most some reverse communication within a subsystem .
This theorem requires that the inputs and outputs form distinct components of a tensor product decomposition of the overall Hilbert space; a requirement which is automatic for finite dimensions. For any local quantum field theory we may rely on the fact that operators with support only outside each others light cones must commute. Thus, locality dictates the existence of the required tensor product structure across a black hole’s horizon Hawking76 (); Braunstein13 (). We may therefore apply the circuit equivalence in Fig. 2 to the black hole evaporation of Fig. 1 to give the structure of an arbitrary unitary black hole evaporation process consistent with local physics (see Fig. 3).
Note that Fig. 3 is not to be interpreted as a spacetime diagram. In particular, we do not require that there is any space-like hypersurface which simultaneously cuts through the subsystems there displayed. For example, we do not require that subsystem all arrives in one block for unitary processing inside the black hole. From this perspective, a quantum circuit is a powerful construct.
Strong subadditivity in Fig. 3 gives , and using the unitary invariance of entropy we have leading to
Note that this inequality involves only correlations between external degrees-of-freedom and hence relates quantities which are, in principle, directly observable and reportable. Combining this with the assumption of complete evaporation, Eq. (3), we find
Non-exotic atmosphere 2.c: We shall take assumption 2.c to be equivalent to 1.b(i), that the exterior should not consist of exotic matter. The holographic entropy bound Bousso99 () shows that an entropy of at most can reside between the causal and stretched horizons, so their distinction has negligible effect on our analysis. For a partially evaporated black hole with Bekenstein Hawking entropy , to ensure a non-exotic atmosphere, Eq. (1) requires . Combining this with Eq. (14) yields
Except for the very earliest stages of evaporation, this result yields a contradiction.
Discussion: Both theorems apply to the behavior of large black holes where General Relativistic reasoning is conventionally expected to hold. Theorem 1, which yields a contradiction from the Page time (PT) onward, ignores the local structure of a black hole’s horizon and suggests that huge thermodynamic entropies reside within the black hole. Theorem 2, which yields a contradiction almost immediately once evaporation has begun, incorporates this local structure and instead suggests that huge entropies reside external to the event horizon.
The cheapest resolution, cut along the lines of Occam’s razor, would be to reject assumption 1.a (2.a). However, for unitarity to be preserved, black hole evaporation cannot stop when some ‘stable remnant’ is reached Hooft (); Bekenstein94 (); Giddings95 (), nor can the black hole interior ‘bud off’ as a baby universe Hooft (); Banks84 (). Any such loss of unitarity would infect almost every other quantum mechanical process Hooft (); Giddings13 (). Unfortunately, as already noted, the firewall paradox has precipitated a crisis for the AdS/CFT correspondence Maldacena13 (), so this route to ensuring unitary black hole evaporation lies on uncertain ground, at least until a suitable dictionary can be found between the CFT and black hole interior states. Indeed, our analysis seems to imply that such a dictionary must be dynamic, varying with the black hole’s age.
If we accept 1.a (2.a), either theorem leaves us with a striking dichotomy. Let us start with the consequences of Theorem 1. We must reject at least one assumption of 1.c or 1.b. To start with, were the accessible dimensionality within the stretched horizon larger than the estimate given by the Bekenstein Hawking entropy, we would be able fill a black hole with more thermodynamic entropy than could be accounted for by the entropy that would eventually appear as radiation. Thus, if assumption 1.c failed to hold the theory of black hole thermodynamics, and possibly thermodynamics itself, could not survive.
By contrast, assumption 1.b, although usually considered a consequence of the Equivalence Principle of General Relativity is in fact nothing more than a boundary condition on the quantum fields at the event horizon; it is well known that different choices of ‘vacuum state’ lead to wildly different behaviors for the energy-momentum tensor there. Splitting 1.b into its components: A failure of 1.b(i), would imply that the exterior must consist of super-entropic exotic matter (such as an atmosphere of microscopic black holes), and so would almost certainly have some observational consequences Telescope (). Finally, a failure of 1.b(ii) would imply that by the PT a black hole’s interior is filled with half of a maximally entangled state.
In 2009 it was noted that maximal entanglement between the black hole interior and the radiation implied an absence of entanglement across the horizon Braunstein13 (): “In an arbitrary system where trans-boundary entanglement has vanished, the quantum field cannot be in or anywhere near its ground state. Applied to black holes, a loss of trans-event horizon entanglement implies fields far from the vacuum state in the vicinity of the event horizon.”
We now extend this reasoning: As there is no entanglement between any pieces of the quantum fields within the black hole, no place within the interior can look like a low-energy vacuum state — like regular spacetime. We might say, in this sense, that from the PT onward there is no spacetime within the black hole.
Next, consider the options left by Theorem 2: to reject at least one of the assumptions 2.c or 2.b. Rejecting 2.c is equivalent to rejecting 1.b(i). Therefore, the only other minimal option (rejecting assumption 2.b) would be to assume that communication from the black hole interior to exterior across the horizon was possible. In particular, one might note that a “tunneling” mechanism has been long anticipated to provide a more powerful explanation for black hole radiation Manas (); Parikh00 (); Braunstein13 (). However, tunneling across the horizon aloneParikh00 () is insufficient, as it still leaves unanswered how the degrees-of-freedom from deep within a black hole manage to (non-locally) reach up to just inside the horizon where they can participate in such tunneling. Our mechanism (below) supporting the firewall phenomenon may explain how this can occur.
Simplifying our assumptions: It turns out that there are several ways of reducing the assumptions needed to obtain Theorem 1. First, we may drop assumption 1.c entirely if we can apply the holographic entropy bound Bousso99 () within a black hole’s horizon. Recall the converse interpretation of this bound which states that the minimal area encompassing a given thermodynamic entropy is four times that entropy (in Planck units). The covariant form of this bound Bousso99 () should apply anywhere (including within a black hole’s horizon). Hence, let be the minimal area encompassing a thermodynamic entropy of . By assumption 1.b, this area must be well within the (surface area of the) horizon, implying that . This result directly contradicts Eqs. (3) and (4). Interestingly, if we conservatively assume that this area is centrally located, so that assumption 1.b holds for as long as possible, then an observer freely falling from infinity will hit the firewall almost immediately upon crossing the horizon.
Finally, let us focus on the onset of the paradox: Consider a black hole, created from matter in a pure quantum state, which unitarily evaporates at least until the Page time. We shall only suppose that the net (von Neumann) entropy in the radiation is equal to the Bekenstein Hawking entropy of the black hole at that stage. We then trivially have . Therefore, if this black hole is to be free of an exotic atmosphere at the Page time, then virtually all this thermodynamic entropy must reside inside, out of view. Consistency with black hole thermodynamics (or the holographic entropy bound) then dictates that there is no room left within the black hole for a visitor to keep her cool!
Speculation: We end with a purely speculative description for how black holes might evolve during evaporation and consequently outline a mechanism supporting the firewall phenomenon, as suggested by our work. During the initial stages of evaporation, prior to the Page time (PT), entanglement grows between the interior and distant radiation. Now entanglement cannot be compressed into fewer qubits than given by its entropy (a principle which in no way assumes that the associated matter has been compressed to Planck densities). Therefore, accepting an effectively finite size Hilbert space to the black hole interior (or applying the holographic entropy bound there), the interior slowly fills up with incompressible entanglement. This incompressible ‘stuff’ would grow outward from what would classically be the black hole singularity, while simultaneously, the black hole’s horizon is shrinking as radiation is emitted.
We conjecture therefore that there is some well-defined internal entanglement surface that contains the entanglement growing outward. At the PT the entanglement surface and black hole horizon meet. At that stage the horizon may survive, or may be replaced by the entanglement surface. In the former case, evaporation would continue by something very much like quantum tunneling Parikh00 (); Braunstein13 (); Manas () from degrees-of-freedom on the entanglement surface just inside the horizon. In the latter case, evaporation may continue via direct ejection from the entanglar (entangled star) though its detailed spectrum (e.g., its neutrino flux) and lifetime would almost certainly differ from a true black hole with otherwise identical mass, charge and angular momentum. Naïvely, an entanglar (of even a modest size) would take far longer than the age of the universe to evolve from a black hole, so none can be expected to currently exist. Conversely, the unambiguous observation of such an entanglar would yield prima facie evidence for an object that far predates the Big Bang.
The ER=EPR ‘counterexample’
Classical 3-Manifold structure: We start by considering the prime counterexample considered in the ER=EPR proposal Maldacena13 (). This consists of a pair of black holes connected by an Einstein Rosen bridge. (Physically, this might correspond to what is produced by a pair-creation event.)
If we ignore evaporation, the black hole exteriors are static and eternal. Their joint Penrose diagram is shown in Fig. 4 as the left and right black diamond shapes of either of the left-hand figures. Each dashed red line denotes some specific space-like hypersurface (a specific time slice). In Fig. 4(a), this time slice is chosen when the two exteriors touch at the center of the left-hand figure. The black diagonal lines passing through the center of this figure represent the horizons of the two black holes respectively. With regard to the scenario where these black holes are created by some pair-creation process, this “” time slice would correspond to their initial creation event, and the Penrose diagram loses any meaning for earlier times. The right-hand diagram is the spatial embedding diagram corresponding to this initial time slice. Far from either black hole, externally, the embedding diagram looks flat. As one approaches either black hole from the outside, one approaches a horn-like structure on the embedding diagram. The horn structure terminates at the horizon, denoted as a vertical black ring that encircles the horn. For the hypersurface the horizons of the two black holes coincide.
Fig. 4(b) shows the same black hole pair, but a later hypersurface is chosen (left-hand figure). The corresponding embedding diagram (right-hand figure) shows that the two horizons have separated and are connected internally by a bridge — the so-called Einstein-Rosen (ER) bridge. The proper length of this bridge grows very rapidly (at roughly the speed of light) so there is no possibility of passing from the exterior of one black hole to the exterior of the other. The pair forms an example of non-traversable wormhole.
Quantum fields and entanglement: Continuing to ignore evaporation, we can consider quantum fields propagating on the time-evolving family of spatial 3-manifolds corresponding to the family of embedding diagrams for different hypersurfaces (time slices). Provided we stay away from the singularity (wavy blue line at the top of the Penrose diagrams) all these 3-manifolds are smooth and locally flat. The lowest energy states (vacuum) of these quantum fields will then not be too different from the structure of vacuum in flat spacetime. In particular, there will be entanglement across all boundaries. This may be formalized, for example on a quantum field theoretic setting on a lattice BDS13 (); Braunstein13 (), with the typical lattice spacing determining the UV cutoff as Planckian. It is sufficient for our purposes here to consider schematic pictures of the entanglement on such a lattice representation of these 3-manifolds.
Fig. 5 shows such a schematic representation of entanglement on these 3-manifolds. The lattice sites are represented as small blue circles. For clarity, only those circles neighboring horizons are shown. Entanglement across the horizons is shown as pale blue lines connecting lattice sites. As can be seen in Fig. 5(a) the initial hypersurface does indeed show entanglement across the joint horizon. The exteriors of the two black holes are indeed entangled on this time slice.
In Fig. 5(b), we see the entanglement for a hypersurface at . Once the horizons have separated by even of order one lattice site, presumed to be separated by , the entanglement between the black hole exteriors includes a set of intermediary lattice sites. When these intermediary sites are traced out the original entanglement will almost have vanished. As the separation between the horizons increases the entanglement between the black holes is exponentially suppressed, effectively vanishing. Thus, on a later hypersurface, such as Fig. 5(c), there will be no entanglement between the black holes.
We may therefore conclude, (i) that on the initial hypersurface there is indeed entanglement of the external degrees-of-freedom for the black hole pair. However, (ii) this vanishes within . Further, (iii) being local entanglement across a horizon this has no observable consequences. For such a short-lived phenomenon one might question whether this entanglement is perhaps better thought of as a mathematical artifact.
Before we proceed to seeing how theorem 1 applies to this counterexample we might consider other ways of creating maximally entangled pairs of black holes. Indeed, Ref. Maldacena13, suggests other mechanisms by which the internal degrees-of-freedom of a pair of black holes may be maximally entangled. For example, by waiting for one black hole to radiate until its Page time and then collapse the resulting radiation into a second black hole. However, all of these alternative suggested mechanisms have entanglement of a completely different character than the counterexample studied above. The entanglement is not ephemeral and it is between the internal instead of exterior degrees-of-freedom. Thus, there is no connection between the smoothness or otherwise of the quantum fields for these mechanisms and the counterexample above.
Theorem 1 for the ER=EPR ‘counterexample’: Finally, we shall consider evaporation in the scenario of pair-created black holes studied above. To apply theorem 1, all be need to do reinterpret , , , etc as the Hilbert spaces of the joint interior, the combined neighborhoods and combined radiation systems for the black hole pair. Assumption 1.c needs to be modified to “the joint black interior Hilbert space dimensionality may be well approximated as the exponential of the combined Bekenstein Hawking entropy of the black hole pair”. As noted in Fig. 1 of the manuscript, the quantum state of is arbitrary. All the equations used to derive the contradiction for theorem 1 remain unchanged. We find the same paradox as before, with its onset at the Page time, when the joint surface area of the black hole pair has dropped to one-half its initial value.
The proposed counterexample to the paradox thus fails.
Post-firewall paradoxes with negligible entropies
In this section we repeat the key elements of both theorems in the manuscript with the following modifications: (a) We explicitly include the entropy in the atmosphere, bounding its size rather than merely considering it to be negligible; (b) We only follow the black hole evaporation to the point where the black hole is still much larger than Planck scale. To illustrate that neither of these changes affect the results of the manuscript we focus solely on the behavior at the Page time.
Consider now the scenario where we follow a black hole to a relatively late stage of its complete evaporation. In particular, when its area has shrunk to some small fraction of its original size, but is still much larger than the Planck scale so that the physics of Planck scale black holes plays no part. We denote all pre-Page time radiation as and the post-Page time radiation as (produced up until the black hole has reached a specified fraction, say roughly , of its original area). It follows therefore from the generic behavior of entropy during evaporation Braunstein13 () that
Combining this with Eq. (2) of the manuscript we find
Equation (10) tells us that the radiation is almost perfectly maximally entangled with a subspace of the joint system and as quickly becomes remotely separated we may conclude that represents a lower bound to the thermodynamic entropy of this joint system.
Free-fall equanimity: Consider now a freely-falling observer who is believed to see nothing special until they pass well within a large black hole’s horizon (assumption 1.b). For black holes formed by matter in a pure quantum state, the (global) state of may also be treated as pure implying . This in turn, allows assumption 1.b to be decomposed into external and internal constraints.
Externally, we assume that our infalling observer is not passing through an atmosphere of exotic matter prior to reaching the horizon. Therefore from Eq. (1) of the manuscript, we have for a black hole at the Page time.
Internally, this implies that Eq. (10) reduces to
Now, a trivial bound to the quantum mutual information is that . If this bound were saturated, the huge thermodynamic entropy inside would imply that an infalling observer would immediately encounter an incredibly mixed state (e.g., a near uniform mixture of roughly orthogonal quantum states for an initially stellar mass black hole) with correspondingly huge energies as soon as they passed the horizon. They would immediately encounter an ‘energetic curtain’ Braunstein13 () or firewall AMPS () upon entering the black hole. To guarantee assumption 1.b holds, the above dimensional bound must be far from saturation, i.e., at the Page time
Finite interior Hilbert space: We may now derive a contradiction along the lines of the original firewall paradox. Assumption 1.c holds that the black hole interior has a Hilbert space dimensionality that is well approximated by the exponential of the Bekenstein-Hawking entropy. Thus, at the Page time, when a black hole’s surface area has shrunk to one-half of its original value we would have
which directly contradicts Eq. (12).
Note that Eq. (6) of the manuscript involves only correlations between external degrees-of-freedom and hence relates quantities which are, in principle, directly observable and reportable. Combining this with the assumption of complete evaporation, Eq. (9), we easily find
Locality (assumption 2.b) has allowed us to eliminate from Eq. (10), which in turn allows us to do without any specific bound to the size of the interior Hilbert space. More surprisingly, locality implies a very different picture: one where huge thermodynamic entropies must reside outside the black hole instead of inside it.
At first sight, this appears reminiscent of arguments based on time-reversing Hawking radiation. Ordinary Hawking radiation evolves out of vacuum modes, but any (information bearing) deviations were argued to have started out as high-energy excitations near the horizon Giddings94 (). By contrast, the huge entropies in Eq. (14) are associated with degrees-of-freedom that are maximally entangled with the outgoing radiation and therefore correspond to an effect of the “infalling partners” to the radiation. Thus, Eq. (14) represents a distinct (and much stronger) phenomenon imposed by locality.
Non-exotic atmosphere: Assumption 1.c is weaker than 1.b, only requiring that externally, black holes should resemble their classical counterparts (aside from their slow evaporation). In turn, we apply this in a weak manner to only suppose that the black hole does not contain an atmosphere of super-entropic exotic matter. From Eq. (1) of the manuscript
whatever the details of the radiation process.
Arbitrary infallen matter
In this section we generalize our results to show that they apply even when the matter that collapsed to form the black hole is not pure. We start with a more general review of generic black hole radiation necessary to analyse such scenarios.
Generics of black hole radiation
In the manuscript we considered a black hole with (initial) thermodynamic entropy which can completely evaporate into a net pure state of radiation. As discussed, the generic evaporative dynamics of such a black hole may be captured by Levy’s lemma for the random sampling of subsystems from an initially pure state consisting of qunats Braunstein13 (). This either assumes the infallen matter is pure (as in the manuscript) or ignores it entirely. Throughout, we set Boltzmann’s constant to unity and work with natural logarithms leading to the measure of qunats (i.e., times the number of qubits bit ()).
In order to extend our analysis to include infallen matter carrying some (von Neumann) entropy , we need only take the initially pure state used above and replace it with a bipartite pure state consisting of two subsystems: qunats to represent the degrees-of-freedom that evaporate away as radiation; and a reference subsystem. Without loss of generality, the matter’s entropy may be treated as entanglement between these two subsystems, however, here we shall simplify our analysis by assuming uniform entanglement between the black hole subsystem and reference qunats. The generic properties of the radiation may then again be studied by random sampling the former subsystem to simulate the production of radiation Braunstein13 ().
The behavior is generic and for our purposes may be summarized in terms of the radiation’s von Neumann entropy, , as a function of the number of qunats in this radiation subsystem. One finds Braunstein13 () that initially increases by one qunat for every extra qunat in , until it contains qunats. From that stage on it decreases by one qunat for every extra qunat in until it drops to when contains qunats and the black hole has completely evaporated.
Because the von Neumann entropy of a randomly selected subsystem only depends on the size of that subsystem, the same behavior is found whether above represents the early or late epoch radiation with respect to any arbitrary split. Further, in the simplest case where we choose the joint radiation to correspond to the net radiation from a completely evaporated black hole we may immediately write down the generic behavior for the quantum mutual information .
In particular, starts from zero when consists of zero qunats. From then on, it increases by two qunats for every extra qunat in until reaches when contains qunats. From that stage on until contains qunats remains constant, after which decreases by two qunats for every extra qunat in until it drops to zero once the contains the full qunats of the completely evaporated black hole Braunstein13 (). Interestingly, it is during the region where is constant that the information about the infallen matter becomes encoded into for the first time Braunstein13 (). Finally, setting to zero gives the ‘standard’ behavior for and upon which the results in the manuscript are derived.
From the above, we are motivated to generalize the Page time: we define any time where is maximal a (generalized) Page time; the earliest such time the ‘initial Page time’; and the latest the ‘final Page time’. Prior to the initial Page time, the quantum information about the initial infallen matter is encoded entirely within the black hole interior Braunstein13 (). After the final Page time this information is encoded entirely within the radiation Braunstein13 ().
Including infallen matter
Let us start with a consideration of how the reasoning in Theorem 2 becomes modified by the presence of infallen matter carrying entropy.
Theorem 2 generalized: In the main body of the manuscript we did not explicitly include entropy associated with infallen matter. Fig. 6 shows the most general scenario. Subsystem denotes the matter that falls into the region surrounding the black hole where radiation is produced. Thus, we suppose that late epoch radiation can in principle come from the joint subsystem . In this figure we also include subsystem denoting matter that has fallen into the region surrounding the black hole at an earlier epoch or indeed matter that may have collapsed to form the black hole in the first place.
As in the manuscript we apply strong subadditivity:
Here, we used the fact that joint subsystems and are unitarily related. Finally, the most natural assumption is that the infallen matter is independent of the quantum state of the black hole, , or its early epoch radiation . The original inequality of Eq. (10) from the manuscript is thus found to still hold in the presence of infallen matter.
From the summary above of generic radiation production including infallen matter we have enough to generalize Theorem 2. As in the manuscript, we take to be all the early epoch radiation until the Page time (for this theorem we may take any generalized Page time), and we let denote all the radiation generated from the Page time onward until the black hole has shrunk to a size much smaller than the original black hole (say roughly of its original area), but still much larger than the Planck scale. In this case, instead of Eq. (9), we have
Once again we obtain a contradiction except in the extreme case of a black hole whose net infallen matter contains virtually as much entropy as the entire black hole’s original entropy .
Theorem 1 generalized: It is simple enough to repeat the above reasoning for Theorem 1, where we no longer make use of locality. In this case, we may still use Fig. 6 provided we ignore the no-communication decomposition structure. In particular, strong subadditivity yields
which is identical to Eq. (2) of the manuscript. Here, we use the fact that joint subsystems and are unitarily related. Again, the most natural assumption is that the infallen matter is independent of the quantum state of the black hole, , or its early epoch radiation .
Applying Eq. (18) to any Page time then tells us that for a unitarily and completely evaporating black hole
To simplify our argument, we shall suppose that the infallen matter ( has actually entered the black hole. In that case, for any times prior to the initial Page time, the infallen matter’s external reference qunats are maximally entangled with some subsystem of the black hole interior Braunstein13 (). We shall label the orthogonal complement of this subsystem within as . It is clear that: i) can be treated as a pure quantum state; and ii) . So that
To ensure that our infalling observer is not passing through an atmosphere of exotic matter before they reach the horizon, Eq. (1) from the manuscript for a large black hole implies that Eq. (22) reduces to
Since by construction, we find the trivial bound
If this bound were saturated, then the huge thermodynamic entropy in would imply that an infalling observer would immediately encounter an incredibly mixed state with correspondingly huge energies as soon as they passed the horizon. They would immediately encounter an ‘energetic curtain’ or firewall upon entering the black hole. To ensure, therefore that assumption 1.b holds, the above bound must be far from saturation, i.e.,
where is the black hole at the initial Page time.
However, assumption 1.c would require that the left- and right-hand-sides of Eq. (25) should be nearly equal. As with the generalization of Theorem 2, we again obtain a contradiction, in this case, however, apparently independent of the amount infallen matter.
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