# The double doors of the horizon

Erik Aurell<EMAIL_ADDRESS>KTH – Royal Institute of Technology, AlbaNova
University Center, SE-106 91 Stockholm, Sweden

###### Abstract

In statistical mechanics entropy is a measure of disorder obeying Boltzmann’s
formula $S=\log{\cal N}$, where ${\cal N}$ is the accessible phase space
volume. In black hole thermodynamics one associates to a black hole an entropy
Bekenstein-Hawking $S_{BH}$. It is well known that $S_{BH}$ is very large for
astrophysical black holes, much larger than any collection of material objects
that could have given rise to the black hole. If $S_{BH}$ is an entropy the
question is thus what is the corresponding ${\cal N}$, and how come this very
large phase space volume is only opened up to the universe by a gravitational
collapse, which from another perspective looks like a massive loss of
possibilities. I advance a hypothesis that the very large increase in entropy
can perhaps be understood as an effect of classical gravity, which eventually
bottoms out when quantum gravity comes into play. I compare and discuss a
selection of the very rich literature around these questions.

###### pacs:

03.67.Lx, 42.50.Dv

Introduction: The Bekenstein-Hawking entropy $S_{BH}$ of a black hole is a
central quantity in modern physics. In its general form $S_{BH}$ equals
$\frac{1}{4}\frac{A}{l_{p}^{2}}$ where $A$ is the surface area of the black
hole horizon, and $l_{p}\approx 1,6\cdot 10^{-35}\,\hbox{m}$ is the Planck
length. This form of the entropy formula includes extreme cases where the
black hole is rotating at high speed and/or is very strongly electrically
charged, and where $A$ can be arbitrarily small. In the simplest setting of a
non-rotating electrically neutral black hole (Schwarzschild geometry) the area
of the horizon is $16\,\pi\,l_{p}^{2}\,\frac{M^{2}}{m_{p}^{2}}$, where $M$ is
the mass of the object and $m_{p}\approx 2\cdot 10^{-8}\,\hbox{kg}$ is the
Planck mass. In this case, roughly appropriate for all black holes that have
been observed to date, $S_{BH}$ equals $4\pi\,\frac{M^{2}}{m_{p}^{2}}$ and
increases faster with mass than any aggregate of particles and fields that
could have formed the black hole Bekenstein (1973); t’Hooft (1993).
Consequently, for large enough mass, $S_{BH}$ will be larger than than the
entropy of any astrophysical object that could have formed the black hole.
Well-known estimates say that the entropy of star like the sun would increase
about $10^{19}$ times if it could collapse into a black hole. In the total
entropy budget of the universe almost all the entropy is held by super-massive
black holes Egan and Lineweaver (2010).

From the point of view of statistical mechanics entropy is a measure of
disorder obeying Boltzmann’s formula $S=\log{\cal N}$. Disorder means that the
system can only be described up to a phase space volume ${\cal N}$. While the
formation of an event horizon closes the door on some types of classical
evolution, since a classical observer can never get out of a black hole, in
some other sense a gravitational collapse must open a door to new
possibilities; ”Opened are the double doors of the horizon // unlocked are its
bolts” Glass et al. (1984).

In this note I put forward a hypothesis that the increase of disorder is due
to the classical chaotic dynamics of a sufficiently strong gravitational
field. The existence of such a mechanism is plausible, and supported by well-
established theories, to be referenced below. The mechanism has nevertheless
not previously been put forward as a physical mechanism which can generate
$S_{BH}$.

The relation to Hawking’s black hole information paradox: Bekenstein-Hawking
entropy is at the center of a very extensive literature motivated by Hawking’s
black hole information paradox Hawking (1976); Preskill (1992); Verlinde and
Verlinde (2013); Harlow (2016); Unruh and Wald (2017). Possible solutions to
the information paradox were in Giddings (1998) classified as (i) fundamental
information loss, (ii) remnants or new physics at the horizon, and (iii)
information return in Hawking radiation. I am here only concerned with the
last scenario, first proposed by Page Page (1980), and more recently dubbed
the ”central dogma” of black hole quantum physics Shaghoulian (2021). Quantum
mechanics is thus assumed to hold everywhere in the universe, including in the
interior of black holes. Some (unknown) unitary operator evolves an initially
pure state of matter to a gravitational collapse and then onto Hawking
radiation escaping to infinity. In Aurell et al. (2021, 2020) we considered
the kinematics of such a process, where in the final state radiation will be
in multi-mode entangled state. That such multi-party entanglement should be
present in Hawking radiation is known already from Page (1980), where it was
pointed out that when one Hawking particle is emitted with some momentum, the
remaining black hole must carry the opposite momentum. Hence later Hawking
particles must be entangled with earlier Hawking particles, though only weakly
by this recoil effect.

A distinction now has to be made between entropy in the informational-
theoretic sense, and entropy in the thermodynamic sense. Entropy of the first
kind, usually called ”fine-grained entropy” in the black hole literature, does
not change under unitary evolution. Entropy of the second kind, usually called
”coarse-grained entropy” in the black hole literature, will however tend to a
maximum at given external constraints. A part of a resolution of the
information paradox of type (iii) must hence be that information-theoretic
entropy is preserved in unitary evolution, and that $S_{BH}$ is a
thermodynamic entropy, an observation also due to Hawking Hawking (2014).
Another argument that $S_{BH}$ is a thermodynamic entropy follows from
comparing $S_{BH}$ to the entropy of black-body radiation emitted by an
evaporating black hole, mode by mode. This can be estimated from the law of
Stefan-Boltzmann and its generalizations Zurek (1982); Page (1983, 2005), and
is somewhat larger than $S_{BH}$, as appropriate for an irreversible process.
The ratio is however a constant independent of mass, meaning that on a
relative scale almost all the entropy increase happens from when matter starts
collapsing until when the resulting black hole has settled down to a
stationary state.

In the quantum domain the role of information-theoretic entropy is taken by
von Neumann entropy, which conserved for a closed system, but can change for a
subsystem. Important progress was in the last less than ten years made in
describing a black hole as a unitary quantum system with
$\exp\left(\frac{A}{4l_{p}^{2}}\right)$ states Lewkowycz and Maldacena (2015);
Engelhardt and Wall (2015); Almheiri et al. (2019); Penington et al. (2019);
Almheiri et al. (2020). This approach allowed for the first time a calculation
of the von Neumann entropy of Hawking radiation, showing that it first
increases, as does the thermodynamic entropy (as above), but then decreases,
following the Page curve. This approach however does not deal with the degrees
of freedom of the space-time geometry in its entirety, including the full
interior of the black hole Maldacena (2021). The question of how thermodynamic
entropy can increase so dramatically in a collapse to a black hole, a process
which can be taken to happen before any Hawking particles have been emitted at
all, has also not been addressed in this important series of papers.

Black holes quantum cores, chaos and space-time chaos: I now introduce three
assumptions of a physical or plausibility nature, patterned after the ”naive
model” of Rovelli and Vidotto (2014), see also Paik (2018): (A) after the
gravitational collapse has run its course there emerges in the center of the
black hole a quantum core state of matter and gravitation; (B) the spatial
support of this state is about $r_{c}\sim
l_{p}\,\left(\frac{M}{m_{p}}\right)^{\frac{1}{3}}$ such that the density of
the core is about Planck density; (C) outside the core space-time is a
meaningful concept, and to a good approximation given by a metric which obeys
Einstein’s equations. All the mass that went down into the black hole is
assumed concentrated in a volume of size $r_{c}$, and thoroughly transformed
by and entangled with the gravitational field. For a solar mass black hole
$r_{c}\sim 10^{13}\,l_{p}\approx 10^{-22}\,\hbox{m}$.

On a general level, during the collapse process the space-time inside a
physical black hole can be assumed to be both unique and complex, where in
classical theory increasing gravity would eventually lead to a singularity in
a way that reflects the black hole’s formation Uggla (2013). One can try to
estimate using classical concepts the increase of disorder when a body falls
down in the black hole before it hits the core. First, it is reasonable to
take any such body an elementary particle, as macroscopic bodies are torn
apart by tidal forces well before hitting the final singularity 111 On a body
of size $l$ and mass $m$ at a distance $r$ from a black hole of mass $M$
($l\ll r$ the tidal forces are about $GmMl/r^{3}$ Imagine a system of size $l$
and binding energy $V$ held together by a force about $V/l$; it is then torn
apart at a distance from the singularity of about $r_{*}\sim
r_{s}\left(\frac{l^{2}}{r_{s}^{2}}\frac{mc^{2}}{V}\right)^{\frac{1}{3}}$. For
a hydrogen atom in a solar mass black hole $r_{*}$ is about one millimeter
($10^{-6}r_{s}$), as follows from $l\sim 10^{-10}$m, $r_{s}\sim 10^{3}$m,
$mc^{2}\sim 10^{9}$eV and $V\sim 13$eV. A proton would similarly be torn apart
at a distance of one nanometer from the singularity ($10^{-12}r_{s}$), as
follows from $l\sim 10^{-15}$m and $V\sim mc^{2}$, though one may imagine that
pulling apart the quarks could also lead to particle production and a
transition to another phase. In any case, both distances are small compared to
$r_{s}$ but very large compared to $r_{c}$.. Second, the same diverging tidal
forces mean that a small difference between a reference geodesics and a
deviation grows without limit as the reference geodesics approaches the
singularity. This type of chaos leads to a growth which is algebraic in the
distance from the singularity at the final time, see Mis  (chap. 32.6) or
Bażański and Jaranowski (1989), which in turn translates to a power-law of the
ratio $\frac{M}{m_{p}}$. The number of bits needed to specify an initial
condition leading to a given final condition is thus logarithmic in
$\frac{M}{m_{p}}$. Chaos of the geodetic motion of the body falling down the
hole can therefore only correspond to a sub-leading fraction of the entropy
increase.

Third, disorder can also emerge from the chaotic non-linear dynamics of the
gravitational field itself. This mechanism (BKL scenario) was first proposed
for homogeneous cosmology Belinskii et al. (1970); Khalatnikov et al. (1985);
Belinskii et al. (1982), and has more recently been conjectured to also
describe aspects of the final stages of the collapse of an astrophysical body
Heinzle et al. (2012). For chaotic dynamical systems (coherent in space) the
increase in entropy per unit time is measured the Kolmogorov-Sinai entropy.
For the homogeneous BKL solution KS-entropy (KS-entropy of the Gauss map) is
known, and is positive Khalatnikov et al. (1985). This should however again
only correspond to a sub-leading fraction of the entropy increase. However, in
BKL, every matter particle loses causal contact with every other matter
particle, a property known as asymptotic silence Andersson et al. (2005).
Therefore, one would not expect the gravitational field around different
points to be synchronized, and the general BKL-like solution has consequently
been postulated to be inhomogeneous (turbulent) with different realizations of
the homogeneous solution around different spatial points Belinskii (1992). If
this mechanism of space-time chaos can produce the increase of phase space
volume quantified Bekenstein-Hawking entropy is hence the question.

Stability and instability of Schwarzschild and geometry: The exterior of the
The Schwarzschild black hole was shown to be linearly stable on the physical
level of rigor in the 1950ies and 1960ies Regge and Wheeler (1957);
Vishveshwara (1968); mathematical proofs are quite recent Dafermos et al.
(2019); Johnson (2019). Very recently a mathematical proof of nonlinear
stability was also presented in Dafermos et al. (2021), unfortunately not very
accessible to physicists 222This so far unpublished monograph is 519 pages in
length. In another line of investigations from the same community, dynamics in
the interior region has also been considered from the rigorous point of view
Dafermos (2005); Dafermos and Luk (2017) Two further papers have been
announced, but have so far not appeared.

This (rigorously established) stability does not exclude that in the interior
there can be the kind of instabilities which in plasma physics and
hydrodynamics are called convective Bers (1975); Huerre and Monkewitz (1985).
Imagine a perturbation of the metric at some distance $r$ from the smeared
singularity. Such a perturbation can decay at the point $r$ where it is
introduced (absolute stability), but at the same time grow, for a finite or
infinite time, at smaller $r$. In a classical model this growth can be
unlimited by moving to ever smaller $r$, but by the assumptions made here it
would stop at $r_{c}$. Unfortunately, the disorder created by such an
instability seems quite hard to estimate.

In a quantum theory of gravity the information would instead be stored in the
state at the smeared singularity. The ensemble describing all such space-
times, assuming they will in the end be thoroughly mixed and indistinguishable
from the outside, must be able to encode all quantum states of all systems
that could have given rise to the black hole of given mass, angular momentum
and electric charge. By one of Bekenstein’s original estimates such an
ensemble has an entropy of the order of $S_{BH}$ Bekenstein (1973). More
refined estimates in the same direction were more recently given by Mukhanov
Mukhanov (2003). Hence, one can at least say that the classical chaos of the
gravitational field should not lead to an entropy increase larger than
$S_{BH}$.

## Acknowledgments

The idea that the increase of entropy in a graviational collapse is related to
classical chaos was raised by Prof Bo Sundborg in discussions with the author.
I thank Angelo Vulpiani for many enlightening discussions, Paweł Horodecki and
Michał Eckstein for a previous collaboration which led to this work, and Claes
Uggla and Ingemar Bengtsson for helpful and constructive remarks. This work
was supported by Swedish Research Council grant 2020-04980.

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## Remarks on classical and quantum entropy increase and thermalization:

The mechanism proposed above is based on the distinction between information-
theoretical entropy and thermodynamic entropy. Thermodynamic entropy can in
turn be interpreted as lack of information (ignorance, or coarse-graining)
Szilard (1925); Mandelbrot (1962); Jaynes (1957a, b), or as a property of an
ensemble describing the outcome of possible (sufficiently simple) experiments
after a process of thermalization Khinchin (1949); Lebowitz (1999).

If and how a closed quantum system thermalizes (or does not) is an important
topic, often referred to as the Eigenvalue Thermalization Hypothesis (ETH)
Srednicki (1994); D’Alessio et al. (2016). In quantum chaos growth of disorder
can often be described by classical dynamics which lead to generation of
smaller and smaller structures in phase space, until quantum effects
eventually take over Berry et al. (1979). The classical picture proposed here
is therefore not necessarily in contradiction with the idea that the quantum
state in the center of a black hole is quantum chaotic and in itself scrambles
information in a fast way Sekino and Susskind (2008). A precise mechanism in
this direction, however assuming the holographic principle, was discussed in
Lowe and Thorlacius (2022).

## Remarks on Bekenstein’s 2001 critique of his 1973 interpretation of BH
entropy

Above I evoked Bekenstein’s argument in his pioneering 1973 paper that
$S_{BH}$ is the number of quantum states of matter than could have formed a
black hole Bekenstein (1973). I thus have to address the counter-argument
advanced by Bekenstein in a 2001 review Bekenstein (2001). Bekenstein there
considered a black hole emitting Hawking radiation and at the same time being
fed by a stream of matter so that its mass, angular momentum and charge stay
constant. Bekenstein formulated his objection (page 8 in Bekenstein (2001)) as
follows:

> [The black hole then] does not change in time, and neither does its entropy.
> But surely the inflowing matter is bringing into the black hole fresh
> quantum states; yet this is not reflected in a growth of $S_{BH}$! […] If we
> continue thinking of the Hawking radiation as originating outside the
> horizon, this does not sound possible.

A part of a counter-argument runs as what Bekenstein outlines is not a state
of thermal equilibrium but a non-equilibrium stationary state (NESS). An
entropy flow can be defined as the expected loss of entropy of the stream of
infalling matter, or as the expected gain of entropy of the outgoing Hawking
radiation: in a stationary state these should match. A mechanism supporting
such an entropy flow was discussed by the author and Michał Eckstein and Paweł
Horodecki in Aurell et al. (2021).

Another part of the counter-argument is that the entropy brought into the
black hole by fresh quantum matter is anyway only an extremely small fraction
of all the entropy in play in the process. The increase of Bekenstein-Hawking
entropy of the black hole when its mass increases by $\Delta m$ is
$8\pi\frac{M}{m_{p}^{2}}\delta m$. This is the same as if an impacting body of
mass $\Delta m$ would carry an entropy per unit mass of $c^{2}/k_{B}T_{H}$
where $T_{H}$ is the Hawking temperature of the black hole. For a solar mass
black hole this is about $10^{47}\cdot\hbox{kg}^{-1}$. By comparison, the
standard molar entropy of water at room temperature is $75.33\hbox{J/(mol K)}$
which means about $3\cdot 10^{26}\cdot\hbox{kg}^{-1}$ in the same units. To
emphasize this discrepancy in entropy per mass of more than twenty orders of
magnitude was the main motivation of this note.