YITP-21-76, IPMU21-0049

aDepartment of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan.
bCenter for Gravitational Physics, Yukawa Institute for Theoretical Physics,
Kyoto University, 606-8502, Kyoto, Japan.

cKavli Institute for the Physics and Mathematics of the Universe (WPI),
The University of Tokyo, Kashiwa, Chiba 277-8583, Japan.

The information loss paradox is usually stated as an incompatibility between
general relativity and quantum mechanics. However, the assumptions leading to
the problem are often overlooked and, in fact, a careful inspection of the
main hypothesises suggests a radical reformulation of the problem. Indeed, we
present a thought experiment involving a black hole that emits radiation and,
independently of the nature of the radiation, we show the existence of an
incompatibility between (i) the validity of the laws of general relativity to
describe infalling matter far from the Planckian regime, and (ii) the so-
called central dogma which states that as seen from an outside observer a
black hole behaves like a quantum system whose number of degrees of freedom is
proportional to the horizon area. We critically revise the standard arguments
in support of the central dogma, and argue that they cannot hold true unless
some new physics is invoked even before reaching Planck scales. This suggests
that the information loss problem, in its current formulation, is not
necessarily related to any loss of information or lack of unitarity.
Therefore, in principle, semiclassical general relativity and quantum
mechanics can be perfectly compatible before reaching the final stage of the
black hole evaporation where, instead, a consistent theory of quantum gravity
is needed to make any prediction.

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###### Contents

  1. 1 Introduction
  2. 2 Is there an information loss paradox?
    1. 2.1 Case A: unitarity problem
    2. 2.2 Case B: entropy problem
    3. 2.3 Case C: No paradox
  3. 3 Standard arguments for the central dogma
  4. 4 General relativity or central dogma?
    1. 4.1 Physical configuration
    2. 4.2 Contradiction between B2 and B3
    3. 4.3 Stronger contradiction
    4. 4.4 Limitations of the standard arguments
  5. 5 On the microscopic interpretation of $A/4G$
  6. 6 Comments on black hole complementarity and firewall
  7. 7 Discussion

##  1 Introduction

Observations of black hole exteriors constitute one of the most successful
confirmation of general-relativity predictions [1]. On the other hand, black
hole interiors are still puzzling and lack a consistent theory capable of
describing them due to the presence of a singularity where Einstein’s theory
is expected to break down [2]. According to classical general relativity, this
would not be a problem for outside observers as a black hole is endowed with a
horizon that hides the singularity. However, at the semiclassical level – i.e.
with matter fields quantized in a classical background – black holes turn out
to be less dark than their classical counterpart, indeed they evaporate by
emitting radiation whose spectrum is approximately thermal [3]. This feature
has puzzled physicists for a long time by raising “apparent” contradictions.
The most famous is the “information loss paradox” according to which an
initial pure state describing collapsing matter would evolve into a final
mixed state after black hole evaporation, thus giving rise to an inconsistency
between general relativity and quantum mechanics as no unitary operator exists
that can evolve pure states into mixed states [4]; see also Ref. [5] for a
pedagogical review.

It should be emphasized that this problem appears because one tries to extend
the validity of the semiclassical approach up to the Planck scale, and
extrapolate physical predictions in regimes where quantum-gravity effects
should not be negligible. In this respect, there is not really a paradox but
it could be just our ignorance about Planck-scale physics that prevents us
from concluding anything definite about what happens to the information during
black hole evaporation [6].

A different version of the paradox involving time scales shorter than black
hole life time was formulated. Indeed, if one makes the “apparently” natural
assumption that as seen from the outside a black hole behaves like a quantum
system whose number of degrees of freedom is given by the horizon area, then
one can show that after some time the von Neumann entropy of Hawking radiation
would exceed the thermodynamic entropy of black hole, which is impossible.
This gives rise to a contradiction even before reaching the end point of the
evaporation when the semiclassical approach should still be valid. Thus, it
was argued that the von Neumann entropy of the Hawking radiation should start
decreasing at roughly half of the black hole life time, and follow the so-
called Page curve consistently with the unitary quantum evolution [7, 8, 9].

In the past decades both the aforementioned starting assumption and the
corresponding alternative version of the information loss problem have been
taken very seriously by a large part of the theoretical physics community. To
emphasize the relevance of the hypothesis on the black hole degrees of freedom
as seen from an outside observer, some authors even referred to it with the
expression “central dogma” [10]. Indeed, lots of efforts both to support the
central dogma and to find a resolution to the induced entropy problem have
been made. In particular, recently a novel computation [11, 12, 13, 14, 15,
16, 17] of the von Neumann entropy of the Hawking radiation by using the Ryu-
Takayanagi entropy formula [18] was shown to be consistent with a unitarity
quantum evolution, and was claimed to be a strong motivation in support of the
central dogma.

In this paper we adopt an open-minded and critical point of view in discussing
whether the current formulation of the information loss paradox is well-posed.
We aim at carefully inspecting all the assumptions that are usually made and
that are claimed to lead to the information loss problem(s). In particular, in
Sec. 2 we review various formulations of the problem by emphasizing some of
the details that are often overlooked, and that are necessary for the
subsequent analysis. In Sec. 3 we examine the standard arguments in favour of
the central dogma. In Sec. 4, we present a thought experiment which unveils an
incompatibility between the validity of semiclassical gravity to describe
infalling matter far from the Planckian regime and the central dogma. This
allows us to argue that as long as the low-energy effective field theory holds
true, i.e. far from the Planck scale, such that the horizon can be described
as a smooth surface, then there exists no information loss paradox. Moreover,
in Sec. 5 we comment on the microscopic interpretation of the Bekenstein-
Hawking entropy; while in Sec. 6 we discuss the relevance of our conclusions
in comparison to the black hole complementarity and the firewall paradigm. In
Sec. 7, we summarize our results and discuss the power of our conclusions.

Throughout the work we adopt the units $c=\hbar=k_{\rm B}=1.$ We will use
different symbols for several entropies: $S_{\rm BH}$ for the Bekenstein-
Hawking entropy; $S_{\rm bh}$ for the total black hole entropy that does not
necessarily coincide with $S_{\rm BH}$; $S_{\rm th}$ for the thermodynamic
entropy; $S_{\rm vN}$ for the von Neumann entropy; $S_{\rm rad}$ for the von
Neumann entropy of the Hawking radiation; $S_{\rm m}$ for the von Neumann
entropy of additional matter.

##  2 Is there an information loss paradox?

It is often stated that the thermal nature of black holes eventually leads to
an incompatibility between semiclassical general relativity and one of the
pillar of quantum mechanics, i.e. the unitarity of the $S$-matrix, thus
leading to the well-known information loss paradox [4]. However, we strongly
believe that the assumptions on which the paradoxical implications rely are
often overlooked, and deserve a deeper and thorough inspection. Therefore, it
is crucial to reformulate the problem by highlighting and critically revising
all the basic assumptions that are usually made. In what follows we carefully
analyze three different scenarios that are normally considered.

###  2.1 Case A: unitarity problem

The simplest formulation of the information loss problem can be stated as the
incompatibility between the two following assumptions.

  1. A1.

Quantum states evolve in a unitary way. In particular, pure states evolve into
pure states;

  2. A2.

Semiclassical general relativity is a valid low-energy effective field theory
to describe black hole physics during the entire evaporation process: black
holes evaporate completely emitting thermal radiation and end up leaving a
regular spacetime (see Fig. 1a).

The hypothesis A1 implies that there exists a unitarity $S$-matrix operator
that describes the evolution from ingoing collapsing matter to outgoing
radiation during the entire black hole formation and evaporation. Whereas A2
means that the semiclassical approach, according to which fields are quantized
in a classical curved background, is always valid even at the end point of the
evaporation.

These two assumptions are clearly incompatible. If the emitted radiation is in
a pure thermal state (as suggested by Hawking’s calculation [3]), then after
the black hole evaporation the final state will be thermal and mixed. This
scenario is illustrated in Fig. 1a, in which it is clear that before reaching
the end point of the evaporation we can draw Cauchy surfaces, e.g.
$\Sigma_{1}$ and $\Sigma_{2},$ on which the state of the joint system (black
hole plus radiation) is pure; whereas after the evaporation the state on each
Cauchy slice turns out to be mixed, e.g. on $\Sigma_{3}.$ Therefore, if the
matter that formed the black hole was initially in a pure state, the previous
argument would imply a breakdown of the unitary condition on the evolution
operator in quantum mechanics as pure states cannot evolve unitarily into
mixed states, thus contradicting A1.

Obviously, there is no reason to trust the semiclassical picture up to the end
of the black hole evaporation, indeed it is quite reasonable to believe that
at least in the latest stages quantum-gravity effects cannot be neglected and
should be consistently taken into account. In fact, it is not clear what the
final state would be. For instance, it could be given by a naked timelike
singularity [3] as depicted in Fig. 1b. If this is the case, we would not be
able to conclude that the final state is mixed. Indeed, $\Sigma_{3}$ in Fig.
1b is not a Cauchy hypersurface, meaning that we would need a description of
the singularity in order to predict the final quantum state.

Furthermore, let us note that both the regular spacetime and the naked
singularity spacetime represent only some arbitrary extrapolation of
semiclassical gravity beyond its regime of validity. In fact, these spacetimes
are not globally hyperbolic and contain a Cauchy horizon $\mathcal{C}$. The
shaded orange regions in Figs. 1a and 1b are in the causal future of the
singularity, and it is not possible to evolve the initial value problem from
the hypersurface $\Sigma_{2}$ to $\Sigma_{3}$ without providing a description
of the singularity which is outside the realm of semiclassical gravity.
Semiclassical gravity can only study the maximally Cauchy development [19]
depicted in Fig. 1c. In this portion of spacetime the initial value problem is
well defined. However, the spacetime is not geodesically complete, therefore
in principle we should not expect a pure state on $\Sigma_{2}$ to evolve into
a pure state on $\Sigma_{3}.$ Although in a different language, similar
considerations were presented in [6].

Hence, this weak formulation of the information loss problem would not be
particularly worrisome. On the other hand, it is well known that one can
formulate a stronger version of the paradox according to which problems seem
to arise even when the black hole mass is much larger than Planck mass.

(a) Subfigure 1 list of figures text

(b) Subfigure 2 list of figures text

(c) Subfigure 3 list of figures text

Figure 1: Penrose diagrams of an evaporating black hole with (a) regular and
(b) singular spacetime (naked timelike singularity) as the final state. In
both cases, the spacetime has a Cauchy horizon labeled by $\mathcal{C}$. If
the final state is singular or we only consider the maximal Cauchy development
(c), a unitary operator does not necessarily evolve a pure state on
$\Sigma_{2}$ into a pure state on $\Sigma_{3}$. The blue dotted line
represents the ($\text{inner}\,\cup\,\text{outer}$) trapping horizon.

###  2.2 Case B: entropy problem

We now present a stronger formulation of the information problem originally
due to Page [7, 8, 9], that it is sometimes referred to as the “entropy
problem”; see also Ref. [10]. It consists in the incompatibility among the
following assumptions.

  1. B1.

Quantum states evolve in a unitary way. In particular, pure states evolve into
pure states;

  2. B2.

Semiclassical general relativity is a valid low-energy effective field theory
to describe black hole physics far from the Planckian regime;

  3. B3.

As seen from the outside, a black hole behaves like a quantum system whose
number of degrees of freedom is given by $A/4G,$ with $A$ being the apparent-
horizon area.

It is clear that in this formulation the first assumption is left unchanged
($\text{A1}=\text{B1}$), whereas the second one has been significantly
weakened ($\text{B2}\subset\text{A2}$). Indeed, B2 means that the
semiclassical approach can be trusted only for black holes whose mass is
sufficiently larger than Planck mass, and the horizon is still assumed to be a
smooth surface, i.e. an infalling observer would experience nothing special
when crossing the horizon, consistently with the equivalence principle.

Furthermore, it is important to note that, if the mass of the black hole is
sufficiently larger than Planck mass, it is possible to foliate the spacetime
in a way that each Cauchy hypersurface intersects all the infalling matter and
outgoing Hawking radiation, but avoids regions with high curvature. Also, the
extrinsic curvature of the hypersurfaces remains small. This foliation is
usually referred to as “nice slicing”. As a consequence of the nice slicing,
if B2 holds, the semiclassical description will be valid also for the matter
and Hawking particles falling into the black hole.

The price to pay in this formulation of the information problem is the
introduction of the third hypothesis B3 that is sometime referred to as
“central dogma” [10], according to which the black hole entropy111Here, we
consider the black hole entropy as the number of physical degrees of freedom
of a black hole as seen from the outside. $S_{\rm bh}$ simply coincides with
the Bekenstein-Hawking entropy $S_{\rm BH}=A/4G,$ i.e. $S_{\rm bh}=S_{\rm BH}$
[20, 21]; in other words the dimension of the black hole Hilbert space is
assumed to be $e^{S_{\rm BH}}.$ We will discuss the motivations and
plausibility of the central dogma in the next section. Before that, let us
quickly repeat the standard argument [8, 9, 10] according to which requiring
the simultaneous validity of assumptions B1, B2, B3 leads to a paradoxical
conclusion.

Let $\mathcal{H}_{\rm bh}$ and $\mathcal{H}_{\rm rad}$ be the Hilbert spaces
of the black hole and of the radiation, respectively. B1 implies that, if the
initial state before black hole formation was pure, then the joint state of
black hole plus radiation, i.e. $\left|\psi\right\rangle\in\mathcal{H}_{\rm
bh}\otimes\mathcal{H}_{\rm rad},$ must also be pure. By tracing over the black
hole degrees of freedom, we can obtain the density matrix for the radiation
subsystem and its von Neumann entropy $S_{\rm rad}$ (which coincides with the
von Neumann entropy of the black hole subsystem). From B3 it follows that
during the evaporation process the thermodynamic black hole entropy $S_{\rm
bh}$ decreases as the horizon area decreases. Whereas the entropy $S_{\rm
rad}$ of the outgoing radiation tends to increase according to Hawking
semiclassical computation which can be trusted as long as B2 is valid. This
means that there exists a time scale222Note that different notions of time are
possible. We will use the time function orthogonal to the Cauchy foliation
$\Sigma_{t}$ (each Cauchy hypersurface is a constant-time hypersurface).
$t_{\rm Page}$ – known as Page time – after which $S_{\rm rad}>A/4G=S_{\rm
BH}=S_{\rm bh};$ see Fig. 2 for an illustration. The last inequality states
that the von Neumann entropy of the Hawking radiation (or, equivalently, of
the black hole subsystem) becomes larger than the number of available degrees
of freedom in the black hole, so that for times $t>t_{\rm Page}$ the joint
state of the black hole coupled to the radiation becomes mixed, thus
contradicting the hypothesis B1 and giving rise to the paradox.

To avoid this problem and preserve unitarity, it was claimed that the entropy
of the Hawking radiation should start decreasing at relatively early times
when $t\sim t_{\rm Page}$ and vanish at the end of the evaporation [8, 9, 10].
It is also sometimes claimed [8, 22] that this might happen thanks to non-
negligible non-thermal corrections arising for times larger than $t_{\rm
Page}.$ In this way the von Neumann entropy of the radiation would purify
consistently with unitarity and follow the so-called Page curve; see Fig. 2. A
price to pay in this scenario is the need for new physics far from the
Planckian regime, i.e. abandoning of B2.

Figure 2: This figure shows the behaviors of Bekenstein-Hawking entropy $A/4G$
(red dashed line), and of the von Neumann entropy of the Hawking radiation
$S_{\rm rad}$ (yellow dashed line) as functions of the time $t$ defining the
foliation of Cauchy hypersurfaces. If the thermodynamic entropy of black hole
$S_{\rm bh}$ is equal to $A/4G$ (i.e. if B3 holds), then for times larger than
the Page time $t_{\rm Page}$ the entropy of radiation exceeds the
thermodynamic one and leads to the information loss problem. In this scenario,
the issue would be solved if and only if the entropy of radiation would follow
the so-called Page curve (blue solid line).

###  2.3 Case C: No paradox

In the Case B we noticed that the assumption B3 plays a crucial role in giving
rise to paradoxical conclusions even when quantum-gravity effects are still
negligible, thus implying either the lack of a unitary evolution or the need
for new physics already at early times when the effective-field-theory
description should still be reliable.

We now discuss a third scenario that is usually less appreciated, in which the
assumption B3 is dropped and the only hypothesises are the following:

  1. C1.

Quantum states evolve in a unitary way. In particular, pure states evolves
into pure states;

  2. C2.

Semiclassical general relativity is a valid low-energy effective field theory
to describe black hole physics far from the Planckian regime.

These two assumptions do not necessarily lead to any information loss paradox!
Since the thermodynamic entropy of the black hole is not constrained to be
given by the horizon area, the outgoing Hawking quanta can always be entangled
with the ingoing ones. In other words, the (negative energy) ingoing flux of
the radiation contributes to the increase of the thermodynamic black hole
entropy, so that although the entropy of the radiation can become larger than
the Bekenstein-Hawking entropy, i.e. $S_{\rm rad}>S_{\rm BH},$ it will never
exceed the total black hole entropy $S_{\rm bh}$, and no entropy paradox
arises.

It is worth mentioning that in analog gravity models for black holes [23]
correlations between ingoing and outgoing quanta can be experimentally
measured [24, 25]. Furthermore, results towards the resolution of the
information problem in this context seem to suggest that the outgoing Hawking
radiation is entangled even after the disappearance of the analog black hole
[26]. The extrapolation of these partial results to gravitational black holes
seems to agree with this third scenario for which B3 does not hold. While it
is true that the analogy is far from perfect, it is also true that the lessons
drawn in this context should not be completely ignored as analog systems
represent the closest we can get to the experimental detection of Hawking
radiation.

Given the validity of the assumptions C1 and C2, we can surely state that the
whole dynamics remains unitary as long as time scales shorter than the black
hole life time are considered. Instead, when approaching the end point of the
evaporation process quantum-gravity effects must be taken consistently into
account, and to do so a full theory of quantum gravity is necessary.

Therefore, it is important to stress that the validity of semiclassical
general relativity before the black hole reaches Planckian size is not
necessarily incompatible with the unitarity of quantum mechanics. It is
important to stress that the information may or may not be lost in the final
stages of the evaporation depending on the physics describing that regime, but
this would depend on the full theory of quantum gravity and goes beyond the
scope of semiclassical gravity. While this should definitely be a well-known
fact, the information loss paradox is often stated as the incompatibility
between C1 and C2, whereas the hypothesis B3 is usually implicitly assumed and
taken for granted.

In fact, it should be emphasized that the information loss problem related to
the growing of the von Neumann entropy is just a consequence of imposing B3 as
a hypothesis. In other words, the imposition of the central dogma implies the
emergence of a paradox which, otherwise, would not arise. With this in mind,
in the next section we carefully and critically inspect the motivations that
are usually given in support of the central dogma.

##  3 Standard arguments for the central dogma

In the previous section we have shown that the requirement of the central
dogma in the Case B, i.e. the imposition of the assumption B3, directly leads
to a paradoxical conclusion. Therefore, we believe it is of crucial importance
to better understand the motivations supporting this hypothesis.

  1. (i)

To our understanding, the main motivation comes from the fact that black holes
obey standard thermodynamic laws [27] with an entropy given by the Bekenstein-
Hawking formula [20, 21]

$S_{\rm bh}=S_{\rm BH}=\frac{A}{4G}.$ (1)

From this observation, and from the fact that for a generic system the
thermodynamic entropy $S_{\rm th}$ imposes an upper bound on the von Neumann
entropy $S_{\rm vN},$ i.e.

$S_{\rm th}\geq S_{\rm vN},$ (2)

one would expect that the apparent-horizon area $A=16\pi G^{2}M^{2}$ has an
intrinsic statistical nature and the total number of internal states might be
bounded by $e^{S_{\rm BH}}.$

  2. (ii)

This picture is reinforced by the imposition of the Bekenstein bound [28],
which states that the entropy of any quantum system localized in a region of
circumferential radius $R$ and of total energy $E$ is bounded as

$S\leq 2\pi ER\,.$ (3)

In the case of a Schwarzschild black hole we have $R=2GM$ and $E=M,$ so that
the previous inequality reads

$S_{\rm bh}\leq\frac{A}{4G}=S_{\rm BH}\,.$ (4)

Therefore, if we assume the Bekenstein bound to be valid, then the Bekenstein-
Hawking entropy is the maximal possible entropy for a black hole as seen from
the outside.

The Bekenstein bound seems to be not applicable when self-gravity effects are
not negligible, e.g. for instance during a gravitational collapse and for
sufficiently large cosmological regions. For this reason, Bousso proposed the
so-called covariant entropy bound [29] whose range of applicability is wider
than Bekenstein bound and reduces to the latter when self-gravity effects are
negligible. For a Schwarzschild black hole the Bousso bound coincides with the
one in Eq. (4).

  3. (iii)

The previous bounds on the entropy seem to suggest an intrinsic connection
between geometry and information. This has brought many people to believe in
the existence of a holographic principle [30, 31] according to which the area
of any hypersurface poses a limit on the information that can be stored within
the adjacent spacetime regions. It is often claimed that holography should be
a property of a consistent theory of quantum gravity [32], and it can be
considered the main modern motivation in support of the central dogma.

  4. (iv)

A more recent argument related to the previous ones comes from a computation
of Hawking radiation that involves the use of the Ryu-Takayanagi entropy
formula [18, 33] in the context of holography and AdS/CFT correspondence [34].
Through such a computation one can reproduce a behavior for the von Neumann
entropy of the radiation that is compatible with the Page curve. This result
was interpreted as a strong indication towards the validity of the central
dogma [10]. It must be also emphasized that this result has been rigorously
obtained in simplified settings like 2D black hole spacetimes [11, 12, 13, 14,
15].

In the next section we will present a thought experiment illustrating a simple
physical configuration in which the central dogma does not hold; in
particular, we will comment on the limitations of the previous arguments.

##  4 General relativity or central dogma?

###  4.1 Physical configuration

Le us assume the hypothesis B2 to be valid and consider a physical setup in
which a spherically symmetric solar mass black hole is radiating Hawking
quanta and, at the same time, also accreting matter. The type of matter
falling into the black hole is in principle completely arbitrary, but we can
surely assume that its quantum state is entangled with a second component, so
that the total state of such additional matter degrees of freedom is
pure333For a different type of matter, e.g. in a mixed state, the following
considerations will be more involved but they can be repeated and shown to be
still valid.. For instance, we can imagine a device that generates two fluxes
of matter entangled with each other: one falls into the black hole, whereas
the other reaches $\mathscr{I}^{+}.$ Therefore, while the total state of
matter is still pure, an observer at infinity would only observe the outgoing
flux of matter and measure a mixed state.

We work in the regime in which the outgoing flux of Hawking radiation is
carrying an energy per unit of time much smaller than the mass of the black
hole, so that the adiabatic approximation is valid and we can implement
Hawking computation to obtain the temperature of the black hole:

$T=\frac{1}{8\pi GM}\,.$ (5)

Then, the mass loss is given by the Stefan–Boltzmann law,

$\frac{{\rm d}M}{{\rm d}v}=-\sigma_{\rm SB}T^{4}A=-\sigma_{\rm
SB}\frac{16\pi}{\left(8\pi\right)^{4}}\frac{1}{G^{2}M^{2}}\,,$ (6)

where $\sigma_{\rm SB}$ is the Stefan–Boltzmann constant and $v$ is the
ingoing Eddington–Finkelstein coordinate which provides a notion of time for
an observer on $\mathscr{I}^{-}$ [19].

Figure 3: Penrose diagram of a black hole whose mass is much larger than
Planck mass and that emits thermal radiation according to semiclassical
general relativity. The blue solid line is a timelike hypersurface on which
one places a device that emits two entangled fluxes of matter (red solid
arrows): one falling into the black hole, while the other reaching
$\mathscr{I}^{+}$. The black hole mass remains constant in time as it is
assumed that the additional energy of the ingoing flux compensates the energy
loss due to Hawking radiation.

Furthermore, the ingoing flux of additional matter can be chosen arbitrarily
but, for simplicity, we assume that it has the same energy as the outgoing
flux of Hawking radiation. In this way, the mass of the black hole remains
constant in time and, as a consequence, the horizon area (or, in other words,
the Bekenstein-Hawking entropy $S_{\rm BH}=A/4G$) remains constant too. The
Penrose diagram for this physical setup is shown in Fig. 3.

We can now consider the entropy of the matter degrees of freedom for an
asymptotic observer. If the semiclassical approach holds true in the regime
under investigation (B2), then the horizon is smooth and we would not expect
any kind of interaction between the outgoing Hawking radiation and the
infalling flux of matter. Thus, the behavior of the von Neumann entropy
$S_{\rm m}$ associated to the additional outgoing (or ingoing) matter is
independent of whether $S_{\rm rad}$ follows Hawking computation or the Page
curve. In fact, in both cases the entropy $S_{\rm m}$ is a monotonically
increasing function of time; see Fig. 4 for a schematic illustration. This
means that we do not need to require B1 to elaborate our argument.

Let us also point out that the states associated to the additional matter
degrees of freedom are prepared outside the black hole, and therefore they can
be engineered in such a way to be and stay orthogonal. This means that there
is no dependency relation among the matter states that could even in principle
decrease the value of the entropy $S_{\rm m}.$

###  4.2 Contradiction between B2 and B3

The important point to notice in the above physical scenario is the following.
If we assume the central dogma to be true, i.e. B3, then after some time
$S_{\rm m}$ would exceed the Bekenstein-Hawking entropy of the black hole and
we would conclude that the total state of the additional matter is not pure
anymore. It is not surprising that we would obtain a similar conclusion as in
the case of the entropy-problem version of the information loss paradox (i.e.
Case B in Sec. 2), where in that case the role of $S_{\rm m}$ is played by
$S_{\rm rad}.$

(a) Subfigure 1 list of figures text

(b) Subfigure 2 list of figures text

Figure 4: Schematic representations of the von Neumann entropies of the
outgoing Hawking radiation $S_{\rm rad}$ (yellow solid line) and of the
additional matter degrees of freedom $S_{\rm m}$ (blue solid line), in
comparison with the gravitational Bekenstein-Hawking entropy $A/4G$ (red
dashed line) which is assumed to be constant in our physical setup. In the
plot (a) $S_{\rm rad}$ is determined by Hawking semiclassical computation;
while in (b) $S_{\rm rad}$ follows the Page curve. In both cases, if we assume
the central dogma to be true, then the number of degrees of freedom in the
black hole is not enough to maintain the entanglement between ingoing and
outgoing matter fluxes. At the same time, the Hawking radiation and the
additional matter cannot be entangled with each other if we assume general
relativity to hold at the horizon scale.

However, in the setup just described everything is under control. Indeed, we
know for sure that the matter degrees of freedom inside the black hole are
entangled with the degrees of freedom outside. In particular, since the pair
of (ingoing and outgoing) fluxes is in an entangled pure state, at any instant
of time (or, equivalently, on any Cauchy hypersurface) the degrees of freedom
of matter (delivered by the ingoing flux) inside the black hole are the same
number as the ones of matter (delivered by the outgoing flux) outside of it,
and they will eventually exceed the maximum limit allowed by the central
dogma, i.e. $S_{\rm m}>A/4G.$ This implies that the central dogma cannot be
satisfied in our physical configuration.

Remarkably, this conclusion does not depend on whether the evaporation process
is unitary or not. Indeed our argument applies to both cases of Figs. 4a and
4b. Thus, without requiring B1, in our thought experiment we showed the
existence of a contradiction between B2 and B3.

We strongly believe that the following fact should be highly emphasized. The
information loss problem is usually stated as an incompatibility between
quantum mechanics and general relativity. However, the apparent issue only
arises because of the imposition of the additional hypothesis B3 that is
usually taken for granted; indeed, the usually claimed inconsistency is
between B1 and $\text{B2}\cup\text{B3}$ [8, 10].

In fact, the information/entropy problem in Case B should be formulated as the
incompatibility between the validity of semiclassical general relativity as a
low-energy effective field theory far from the Planckian regime (implying a
smooth horizon) and the central dogma. Therefore, we need to abandon either B2
or B3. In this respect, there exists no information loss paradox since either
$\text{B1}\cup\text{B2}$ (Case C) or $\text{B1}\cup\text{B3}$ does not
necessarily lead to paradox. In our opinion the assumption B3 should be
critically revisited, and indeed the conclusion drawn from our thought
experiment can be also phrased as a counterexample to the central dogma if B2
is assumed to be the valid one.

Obviously, our argument would fail if some new physics at the horizon scale is
invoked. However, since we expect quantum-gravity effects to emerge only when
approaching the end of the evaporation, i.e. at Planck scales, then the
effective-field-theory description is completely valid in the regime we
considered. This is exactly the point that should be stressed. Contrary to
what is usually stated, if no new physics is assumed at the horizon scale then
surely one needs to abandon B3 (i.e. to consider Case C) and then there is no
information loss problem before reaching the latest stages of black hole
evaporation where, instead, a consistent theory of quantum gravity is needed
to make any prediction.

Finally, let us note that a physical setup with infalling additional matter
similar to the one considered in our thought experiment was studied in [35].
The author argues that this configuration does not need to represent a
counterexample to the central dogma if the degrees of freedom of the matter
reach the singularity and are non-locally transferred to the radiation. This
would lead to entanglement between the matter outside the black hole and the
outgoing Hawking radiation. While this is certainly a possibility, it must be
noted that the singularity is in the future of any observer. Therefore, the
mechanism that would be necessary to transfer information from the singularity
to the exterior would not only require interaction to propagate in a spacelike
way (as pointed out in [35]), but also backward in time. Furthermore, in our
scenario this would require the outgoing matter to be maximally entangled
simultaneously with the ingoing matter and the outgoing Hawking radiation,
thus violating the monogamy of entanglement theorem [36]. Therefore, we
believe that this is far from the conservative scenario as it would surely
require new physics beyond the standard semiclassical description, thus
violating B2.

Let us also mention that in the context of quantum teleportation the
information about a given quantum state can be sent combining a purely
classical channel and a purely quantum one [37] and that the amount of
violation of semiclassical causality might be significantly reduced [38].
Although the classical channel can in principle contain very little
information, the quantum channel cannot be decoded by its own without the
classical channel. Therefore, a mechanism that is able to teleport the quantum
portion of the state would not give rise to causality violation. This
partially alleviates the problem discussed in this section as all or a part of
the information encoded in the quantum channel can leak out from the black
hole without giving rise to any causality problem. However, unless at some
point the classical channel is also transferred, the information in the
quantum channel is useless. Invoking the transfer of the classical channel
from the interior of a black hole to the exterior then requires violation of
the semiclassical causality. For this reason, the full information can be
recovered in accordance with the central dogma only if B2 is violated at least
to the extent that allows for the transfer of the minimal amount of
information in the classical channel.

###  4.3 Stronger contradiction

Through the thought experiment described in the previous subsection we argued
that the entropy problem in Case B should be reformulated as a contradiction
between B2 and B3 independently of B1, instead of an incompatibility between
B1 and $\text{B2}\cup\text{B3},$ contrarily to what is often claimed.

Before continuing our analysis, let us note that the assumption B2 can be
split in two parts B1a and B2b such that
$\text{B2}=\text{B2a}\,\cup\,\text{B2b}:$444It is worth mentioning that our
assumptions B2a, B2b and B3 coincide with the postulate 2, 4, 3 in [39],
respectively; whereas our assumption B1 differs substantially from postulate 1
of [39] as we do not assume that the Hawking radiation must purify. Moreover,
in [39] no contradiction between B2b and B3 was noticed; in fact, B3 was
implicitly assumed to be true regardless of the validity of B2b.

  1. B2a.

Black holes whose mass is larger than Planck mass emit thermal radiation
according to semiclassical general relativity;

  2. B2b.

Infalling matter far from the Planckian regime obeys the laws of general
relativity.

The hypothesis B2a means that the semiclassical approach can be trusted
outside the horizon for time scales shorter than black hole life time.
Whereas, B2b means that in the presence of a black hole emitting radiation, an
infalling matter far from the Planckian regime would experience nothing
special, e.g. when crossing and leaving the horizon in the past, consistently
with the equivalence principle, so that the horizon is assumed to be a smooth
surface.

The contradiction we found in our thought experiment is between B2b and B3,
that is, between the laws of general relativity governing the physics of
infalling matter and the central dogma; while B2a and B3 can in principle be
perfectly compatible. This means that the contradiction is even stronger.

On the other hand, one could criticize our conclusion by saying that to
formulate the paradox in Case B only B2a is really needed, and the
incompatibility is between B1 and $\text{B2a}\cup\text{B3},$ independently of
B2b. Although assuming B2a without B2b might require some discontinuous jump
in the physics governing the proximity of the horizon, we agree that such a
logic can be correct and can lead to an information loss paradox. However, we
should immediately notice that this does not imply an incompatibility between
general relativity and quantum mechanics since abandoning B2b already implies
something beyond general relativity.

Indeed, in our thought experiment we clearly showed the existence of a
contradiction between the assumptions B2b and B3, from which it follows that
if B3 is valid, then the infalling matter should experience some new physics
whose description goes beyond the laws of general relativity. Remarkably, this
implies that the information loss problem in the Case B (with B1, B2a, B3) is
a statement of an inconsistency between quantum mechanics and some new unknown
gravitational physics incompatible with general relativity. In other words,
this version of Case B does not show any contradiction between general
relativity and quantum mechanics.

Furthermore, if we assume B2b, then we must drop B3 and we fall into Case C
from which it is clear that no information paradox arises before reaching the
Planck scale where, instead, a consistent theory of quantum gravity is needed
to make any prediction about the black hole final state. Therefore, we should
say that, in the case B, the real incompatibility is between general
relativity and the information loss paradox itself.

###  4.4 Limitations of the standard arguments

One way to interpret the conclusion reached in our thought experiment is that
we provided a simple working example in which the central dogma does not hold
(if B2b is true). Let us now analyze where the usual arguments in favor of the
central dogma reviewed in Sec. 3 fail.

  1. (i)

The standard thermodynamic argument based on the area-law for the entropy of a
black hole fails because it does not take into account the contribution due to
the matter fields inside the black hole, e.g. $S_{\rm m}$ in our thought
experiment. Indeed, it should be clarified that the black hole thermodynamic
laws [27] are intrinsically gravitational in nature. In other words, the
Bekenstein-Hawking entropy $S_{\rm BH}$ takes into account only gravitational
degrees of freedom, and in general the total black hole entropy is given by
$S_{\rm bh}=S_{\rm BH}+S_{\rm m}.$

To understand why additional matter degrees of freedom must be taken into
account for the correct evaluation of the von Neumann entropy, despite they do
not appear in the gravitational thermodynamic laws (thus apparently violating
the inequality (2)), it is useful to consider a system composed by two
subsystems “$A$” and “$B$”. If these two subsystems are very far apart, it is
possible to study the thermodynamic laws of subsystem $A$ without taking into
consideration subsystem $B$. However, this is no longer true if we are
interested in the evaluation of the von Neumann entropy, indeed in this case
we need to consider both subsystems even if they are very far apart because of
quantum correlations due to entanglement. Eq. (2) is only valid if we consider
the thermodynamic entropy of the whole system $A\cup B,$ i.e.

$S_{\rm th}(A\cup B)\geq S_{\rm vN}(A\cup B)\,,$ (7)

but obviously it does not hold if we consider only the thermodynamic entropy
of the subsystem $A$, i.e.

$S_{\rm th}(A)\ngeq S_{\rm vN}(A\cup B)\,.$ (8)

While this consideration may seem obvious, it is at the root of the
contradiction reached in our thought experiment. The role of subsystem $A$ is
played by the gravitational degrees of freedom, whereas subsystem $B$ is given
by the matter degrees of freedom that fall into the black hole. An outside
observer cannot be influenced by such matter degrees of freedom, thus they do
not need to be included in the description of black hole thermodynamics.
However, if these matter degrees of freedom are entangled with degrees of
freedom outside, as it happens in our setup and with Hawking radiation, then
we cannot discard them when computing the von Neumann entropy. In particular,
the fact that at the Page time the entropy of the matter outside (or inside)
the black hole exceeds the area is not in contradiction with Eq. (7), but it
simply confirms that the apparent-horizon area only accounts for the entropy
associated to the gravitational degrees of freedom. Thus, the inequality

$S_{\rm vN}\left(\text{gravity}\cup\text{matter}\right)>\frac{A}{4G}$ (9)

for some states does not lead to any conceptual problem, rather it implies

$\frac{A}{4G}=S_{\rm th}\left(\text{gravity}\right)\neq S_{\rm
th}\left(\text{gravity}\cup\text{matter}\right).$ (10)

Therefore, a priori there is no reason why the addition of matter degrees of
freedom should not increase the entropy of the black hole, especially if B2 is
valid. The same conclusion would also apply to the ingoing Hawking flux which
should contribute to an increase of the thermodynamic entropy of the entire
black hole system made up of gravitational plus matter (Hawking quanta)
degrees of freedom.

The possibility that the Bekenstein-Hawking entropy only accounts for the
degrees of freedom that are accessible (i.e. causally connected) to an outside
observer was already suggested in [40]. Our thought experiment allow us to
conclude that this must be the case if the assumption B2b is satisfied.

This analysis can also easily explain the apparent contradiction between the
Bekenstein-Hawking formula and the fact that “bag of gold” spacetimes [41] can
have a huge volume and a very small area. The area law simply does not
describe all the bulk degrees of freedom inside a black hole. See also Refs.
[42, 43] for a complementary argument against the central dogma.

  2. (ii)

It is often stated that the ingoing flux of Hawking radiation cannot increase
the entropy of the black hole otherwise the Bekenstein bound in Eqs. (3) and
(4) would be violated. However, it must be emphasized that the standard
Bekenstein bound cannot be naively applied when both negative and positive
energies are involved, and it is not clear whether gravitational corrections
can appear when the curved nature of spacetime is taken into account. In fact,
a version of such a bound has been rigorously proven only in the context of
quantum field theory in flat spacetime [44]. To be more precise, although the
proved inequality is applicable to any quantum systems and to any quantum
states, it can be interpreted as Bekenstein bound only in special cases.

From these last observations, we also see no reason why the infalling matter
in our thought experiment should not contribute to the increase of the black
hole entropy and not violate the Bekenstein bound. Moreover, the Bousso
entropy bound, that for a black hole implies the one in Eq. (4), would also be
violated. Indeed, both the statement of the Bousso bound (the covariant
entropy conjecture) and its spacelike projection assume the dominant energy
condition [29] that does not hold in quantum field theory in curved spacetime.

  3. (iii)

As we mentioned in Sec. 3, holography is one of the main motivation that is
usually advocated in support of the central dogma. However, our thought
experiment simply reveals an incompatibility between B2b and B3, where the
latter is normally thought to be a necessity in presence of holography.
Therefore, from our argument it follows that holography, if it implies B3, and
the validity of semiclassical gravity for matter infalling to a black hole
would be incompatible.

We are not claiming that holography is wrong but argued that it is
incompatible with the low-energy effective-field-theory approach according to
which the horizon is smooth as long as the black hole mass is larger than
Planck mass. For instance, quantum-gravity theories that predict new physics
at the horizon scale and that invalidate B2b might be consistent with a
holographic principle [32]. However, we do not see any strong reason why
quantum gravity should unconditionally incorporate or predict holography.
There may be a condition under which holography holds and such a condition may
be violated in some cases. Indeed, it is worth mentioning that several recent
approaches [45, 46, 47, 48, 49, 50, 51, 52, 53, 54] aimed at formulating a
renonormalizable and unitary quantum field theory of the gravitational
interaction seem to be valid proposals for a consistent quantum-gravity theory
that does not rely on or have any relation with holography.

  4. (iv)

Very recently, many interesting works have been done towards the resolution of
the entropy problem outlined in Case B [11, 12, 13, 14, 15]. Although we
appreciate that rigorous computations of the Page curve have been made in some
specific simplified settings, we have reservations about the claimed
interpretation of the result. Let us briefly review the logic that was
followed.

The computation of the von Neumann entropy of the radiation was obtained by
using the Ryu-Takayanagi formula [18] that can be derived from the
gravitational path integral under the assumption of holography [33, 55]. As a
result the Page curve was reproduced, and several authors [11, 12, 13, 14, 15]
claimed that the entropy problem (paradox) was solved. Let us also emphasize
that the reason why the final state of the radiation purifies in their
calculation is that the Ryu-Takayanagi formula for the entropy of the
radiation also includes the contribution from the black hole interior. This
approach to compute entropy of the radiation is also known as “island program”
[10].

One may now wonder what would be the outcome of the island computation when
applied to the von Neumann entropies of radiation and additional matter in our
thought experiment. In fact, if one assumes that the Hilbert space of the
interior of the black hole (including the additional matter) is no longer
independent of the Hilbert space of outgoing radiation, then by using the
island formula one could show that eventually the quantity $S_{\rm rad}+S_{\rm
m}$ will tend to $A/4G$ and never exceed it. If this was the case, then one
would conclude that the central dogma is still respected.

We agree with this possibility; however, we believe that such a scenario is
completely consistent with the conclusion drawn from our thought experiment.
The assumption B3 of central dogma can be satisfied only if B2b is violated.
Indeed, in our physical setup the island would correspond to the entire black
hole interior, and it so happens that the von Neumann entropy $S_{\rm m}$ of
the infalling matter does not contribute to the von Neumann entropy of the
black hole because both ingoing and outgoing fluxes of matter are taken to be
part of the same $\text{island}\cup\text{radiation}$ region. This fact
implicitly assumes that the additional matter and the radiation get somehow
entangled, but this can happen if and only if the matter in free fall
interacts with the Hawking radiation, which would be impossible if the laws of
general relativity are valid, i.e. if B2b holds.

Therefore, when applied to our physical setup the island program implies that
the laws of semiclassical general relativity to describe infalling matter are
violated. This might also suggest that new physics beyond general relativity
is needed to completely trust and physically interpret the island
computation555Also, in our opinion it is not totally clear which Euclidean
solutions should contribute to the island computation based on the replica
method as saddle points, without additional information or assumptions.. Yet
another possible interpretation is that the island formula somehow takes into
account the transfer of information through the quantum channel in the
language of quantum teleportation. However, as explained in the last paragraph
of subsection 4.2, one still needs to transfer the classical channel from the
black hole interior to the exterior and thus the semiclassical general
relativity and the central dogma are incompatible with each other.

See also Ref. [56] where it was recently argued that the island
program/holography might not be consistent with massless gravitational
theories, and Ref. [57] where it was claimed that the transfer of information
based on the island computation requires a nonlocal interaction term in the
Hamiltonian. On the other hand, it is also worth mentioning that other authors
[55] claim that holography must be imposed even when performing the standard
Euclidean computation [58] of the Gibbons-Hawking entropy, or in other words
that the Euclidean gravitational path integral is only valid for holographic
theories.

##  5 On the microscopic interpretation of $A/4G$

Let us now assume that general relativity is a valid theory far from the
Planckian regime, so that the horizon is a smooth surface and a free falling
observer does not experience anything out of the ordinary before approaching
the singularity. Consequently, the central dogma (B3) is violated, and we
would fall into Case C discussed in Sec. 2.3 according to which no information
paradox exists. In such a case, a very natural question to ask is – what is
the microscopic interpretation of the Bekenstein-Hawking entropy $A/4G?$

Macroscopically, i.e. in a coarse-grained sense, $A/4G$ can be interpreted as
the contribution of a black hole to the total thermodynamic entropy that
follows the generalized second law in semiclassical regimes. However, since
the central dogma does not hold, then one should give an alternative
interpretation of the Bekenstein-Hawking formula at microscopic level, i.e.
one should explain which part of the black hole degrees of freedom are
described by $A/4G.$ or instance, in Ref. [59] it was assumed that the
dynamical degrees of freedom of the black hole correspond to thermally excited
modes behind the horizon that are invisible to a distant observer, and it was
shown that their contribution is indeed proportional to the horizon area.

One expects the microscopic description to be highly model-dependent, and that
the correct origin and counting of both gravitational and matter degrees of
freedom can be given by a consistent theory of quantum gravity. In what
follows we address the above question in three different well-known
theories/scenarios.

#### String theory:

In Ref. [60] it was shown that the counting of D-brane states correctly
reproduces the formula $A/4G$ for a class of five-dimensional extremal black
holes. In particular, no more entropy than $A/4G$ can be added to the D-brane,
which is supposed to correspond to a black hole. Furthermore, such
correspondence may be more general and applicable to realistic non-extremal
black holes: the gravitational radius of a D-brane increases as the string
coupling is raised; and the D-brane becomes a black hole when the
gravitational radius becomes as large as the size of a string. If the
discontinuity in the mass at the transition is not too large and if the
typical D-brane states become the typical black hole states then the entropy
of the D-brane correctly accounts for the entropy of the black hole up to a
numerical factor of order unity [61]. Then, apparently, the black hole entropy
cannot significantly exceed the Bekenstein-Hawking value. However, even if the
computation can be extended to realistic non-extremal black holes, we should
notice that it is not known which degrees of freedom in the black hole
spacetime correspond to the D-brane; they may be degrees of freedom localized
at the horizon, those localized slightly inside or outside the horizon, those
localized near the singularity of the classical black hole solution, or
something else. If the D-brane corresponds to degrees of freedom localized or
extended outside the horizon in the black hole spacetime then B2b may be
violated. Whereas, if the D-brane corresponds to some degrees of freedom
localized somewhere strictly inside the horizon (e.g. near the classical
singularity), then additional entropy may be stored in a region inside the
horizon but away from the place where those degrees of freedom corresponding
to the D-brane are localized. In this case, one might argue that the D-brane
states would correspond to the gravitational degrees of freedom, i.e. $A/4G$,
and that the additional states would correspond to the matter degrees of
freedom, i.e. $S_{\rm m}.$ Thus in this case, consistently with our thought
experiment, the total entropy of the black hole would be given by $S_{\rm
bh}=A/4G+S_{\rm m}>A/4G$. In summary the D-brane picture of black holes in the
context of string theory has not yet been fully understood and therefore it is
at this stage compatible with various possibilities, such as the one suggested
by our thought experiment.

#### Loop quantum gravity:

In Ref. [62] the Bekenstein-Hawking formula for the entropy was computed in
the non-perturbative framework of loop quantum gravity. In particular, by
making a quantized phase-space description for the black hole, it was shown
that the resulting statistical mechanical entropy is proportional to the
horizon area; see also Refs. [63, 64] for related works. The important point
to notice in the derivation is that the computed entropy only accounts for the
quantum states that describe the horizon geometry, as also pointed out in Ref.
[62]. Therefore, also in this approach the horizon area counts purely
gravitational degrees of freedom; whereas, matter degrees of freedom are
expected to give an additional contribution to the total entropy of the black
hole (as seen from outside).

#### Corpuscular gravity:

According to the corpuscular picture [65, 66, 67, 68], a black hole can be
described as a self-sustained system of weakly interacting gravitons. In this
case, the number of gravitons $N$ coincides with the Bekenstein-Hawking
formula, i.e. $N\simeq A/4G.$ Therefore, the gravitons microscopically count
the gravitational degrees of freedom of the black hole. Whereas, if we also
include matter in the system, then the corresponding degrees of freedom are
associated with an additional (non-gravitational) entropy contribution, i.e.
$S_{\rm m},$ so that the total black hole entropy is given by $S_{\rm
bh}\simeq N+S_{\rm m}>A/4G.$ It is worth mentioning that, strictly speaking,
for a corpuscular black hole no geometric notion of horizon exists so that, in
general, also B2b is violated. However, one can show that in the $N\gg 1$
limit the corpuscular corrections are negligible, and the values of all
relevant physical quantities are compatible with the presence of a smooth
horizon [65].

##  6 Comments on black hole complementarity and firewall

Let us now make some comments on black hole complementarity [69] and firewall
paradigm [39]. In both cases, the central dogma is assumed as a postulate and
never questioned.

#### Black hole complementarity:

According to black hole complementarity [69, 70], an outside observer can
replace the black hole in terms of a hot membrane whose surface lays one
Planck length above the horizon at the so-called stretched horizon, and whose
entropy coincides with the Bekenstein-Hawking entropy $A/4G$ in agreement with
the central dogma. Whereas, an infalling observer would not see any membrane
in the proximity of the horizon, and would not experience any deviation from
general relativity.

However, from the conclusion of our thought experiment, it follows that the
entropy of the stretched horizon (i.e. of the black hole) can be bounded by
the area only if some new physics beyond general relativity is invoked at the
horizon scale; otherwise an outside observer can prepare an experiment and
confirm that the actual entropy is larger than $S_{\rm BH}.$ Therefore, the
postulates of black hole complementarity are not compatible with a smooth
horizon.

#### Firewall:

The firewall paradigm [39] also states that black hole complementarity is not
compatible with a smooth horizon. However, we should stress that our logic is
completely different from the firewall one. Indeed, in the firewall proposal
the central dogma was imposed as a postulate and, moreover, the state of
Hawking radiation was assumed to purify. In our thought experiment we have not
made any assumption on the final state of the Hawking radiation; in fact, our
conclusions are more general. We showed the existence of an incompatibility
between the validity of general relativity for infalling matter and the
central dogma, independently of whether the von Neumann entropy of the Hawking
radiation follows the Page curve or not.

Furthermore, regarding the question of whether a firewall exists or not, one
cannot really answer without having a consistent theory of quantum gravity
which would allow us to predict the final quantum state of black hole
evaporation. However, we do not see a strong reason why the state of the
Hawking radiation should purify. For instance, if we abandon the central
dogma, then it is also very plausible that the final black hole state is a
remnant, so that the Hawking radiation remains in a mixed state but the total
state of the joint system is pure; see also Ref. [40] for a similar argument.

##  7 Discussion

In this paper, we critically inspected the assumptions behind the
formulation(s) of the information loss problem. Although it is often stated as
an incompatibility between general relativity and the unitarity of quantum
mechanics, we have argued that this is true only if the semiclassical gravity
description is naively assumed to be valid all the way up to the Planck scale,
and if the spacetime is extended beyond the Cauchy horizon in a specific way
(Case A). Therefore, this version of the paradox is not particularly worrisome
as quantum-gravity effects are expected to become relevant at the final stages
of the black hole evaporation.

However, an “apparent” paradoxical conclusion involving general relativity and
quantum mechanics is claimed to arise in regimes where the semiclassical
approach is still valid, once the so-called central dogma is taken as an
additional hypothesis (Case B).

We questioned the well-posedness of this second formulation of the paradox.
Remarkably, by working in a well-defined physical setup consisting of a black
hole whose energy loss due to radiation emission is compensated by an
additional infalling flux of matter, we provided a clear working example to
show the existence of an incompatibility between the following two
assumptions:

  * B2b.

Infalling matter far from the Planckian regime obeys the laws of general
relativity;

  * B3.

As seen from the outside, a black hole behaves like a quantum system whose
number of degrees of freedom is given by $A/4G,$ with $A$ being the apparent-
horizon area.

As a consequence, we argued that Case B does not imply any contradiction
between general relativity and quantum mechanics. Indeed, if the entropy
problem is formulated as in Sec. 2.2, then the incompatibility is between the
validity of semiclassical gravity far from the Planckian regime and the
central dogma, independently of the assumption of unitary evolution, rather
than between the assumption of unitary evolution and semiclassical gravity,
contrarily to what it is often claimed [10].

On the other hand, if the semiclassical gravity assumption B2 is split into
B2a and B2b as described in Sec. 4.3, and if we abandon B2b, then the paradox
can be reformulated as an incompatibility among B1, B2a and B3. However, this
reformulation must require the violation of B2b, which means that it is
automatically inconsistent with the laws of general relativity to describe
infalling matter. Therefore, the paradox would consist in an incompatibility
between quantum mechanics and some new unknown gravitational physics at the
horizon scale. Furthermore, it seems at the very least unnatural to assume the
validity of semiclassical gravity outside the horizon while simultaneously
violating general relativity at the horizon scale.

We are now in a position to address the two questions raised in the titles of
Sec. 2 and Sec. 4.

  1. 1.

Is there an information loss paradox?
Very interestingly, our conclusions imply that the usually stated information
loss paradox has nothing to do with the loss of information, as quantum
mechanics and semiclassical general relativity can be perfectly compatible
before reaching Planck scales where, instead, a consistent theory of quantum
gravity is needed to make any prediction about the final state of black hole
evaporation. A contradiction only arises once the central dogma (B3) is added
to the picture. The main message of this paper is that the problem is not due
to an incompatibility between general relativity and quantum mechanics but,
instead, it is due to a contradiction between B2b and B3 (and so between B2
and B3) independently of B1. To be fair, we should say that, in the case B,
the real incompatibility is between general relativity and the information
loss paradox itself.

Information may or may not be lost due to the formation of an event horizon
[71]. However, here we have shown that there is no reason to argue in any
direction within the regime of validity of semiclassical gravity.

  2. 2.

General relativity or central dogma?
While it is crucial to properly formulate the problem, the question of which
of the two assumptions (B2b or B3) should be dropped cannot be answered. In
fact, there are a number of approaches that aim to address the process of
black hole evaporation; for an incomplete list see [10, 11, 12, 13, 14, 15,
16, 65, 66, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83] and
references therein. However, we should emphasize that it is impossible to
understand whether one of these possibilities is indeed physically correct as
it is currently impossible to perform an experiment to observe Hawking
evaporation. Therefore, the reader is free to choose which of the two
assumptions to save, but at most one.

### Acknowledgements

The authors would like to thank Gia Dvali, Valeri Frolov, Stefano Liberati,
and Tadashi Takayanagi for critical comments and useful suggestions on an
initial draft of this manuscript. In particular, they are grateful to Tadashi
Takayanagi for enlightening discussions. F. D. F. would like to thank Ibrahim
Akal, Raúl Carballo-Rubio, and Mohammad Ali Gorji for several stimulating
discussions. L. B. is grateful to YITP for the warm hospitality, where this
work was started. L. B. acknowledges financial support from JSPS and KAKENHI
Grant-in-Aid for Scientific Research No. JP19F19324. The work of F. D. F. was
supported by Japan Society for the Promotion of Science Grant-in-Aid for
Scientific Research No. 17H06359. The work of S. M. was supported in part by
Japan Society for the Promotion of Science Grants-in-Aid for Scientific
Research No. 17H02890, No. 17H06359, and by World Premier International
Research Center Initiative, MEXT, Japan.

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