# Planck stars, White Holes, Remnants and Planck-mass quasi-particles
The quantum gravity phase in black holes’ evolution and its manifestations

Carlo Rovelliabcd, Francesca Vidottodef a Aix-Marseille University, Université
de Toulon, CPT-CNRS, 13288 Marseille, France.  b Perimeter Institute, N2L2Y5
Waterloo, Ontario, Canada  c Santa Fe Institute, 87501 Santa Fe, New Mexico,
USA  d Philosophy Department and Rotman Institute, Western University, N6A5B7
London, Ontario, Canada  e Physics and Astronomy Department, Western
University, N6A5B7 London, Ontario, Canada  f Instituto de Estructura de la
Materia, IEM-CSIC, Serrano 121, 28006 Madrid, Spain

###### Abstract

This is a review of some recent developments on quantum gravity aspects of
black hole physics. In particular, we focus on a scenario leading to the
prediction of the existence of a Planck-mass quasi-stable object, that could
form a component of dark matter.

###### Contents

  1. I Introduction
    1. I.1 A sketch of the scenario
    2. I.2 The domain of validity of classical gravity
  2. II Non dissipative aspects of the transition
    1. II.1 Planck stars
    2. II.2 Black-to-white transition
    3. II.3 The exterior metric
    4. II.4 The Boundary Region
    5. II.5 The LQG transition amplitude
    6. II.6 White holes
  3. III Dissipative aspects of the transition
    1. III.1 Black hole lifetime
    2. III.2 How big is the interior of a black hole?
    3. III.3 Remnants and their lifetime
    4. III.4 Instability
    5. III.5 Planckian Remnants
    6. III.6 Remnants and quantum spread of the extrinsic curvature
    7. III.7 There is no information paradox
  4. IV Elements of Phenomenology
    1. IV.1 Dark Matter
    2. IV.2 Direct detection
    3. IV.3 Cosmological implications
    4. IV.4 Erebons
    5. IV.5 Modeling remnants emission

##  I Introduction

Quantum gravity is a theory with a mass scale: $m_{P}=\sqrt{c\hbar/G}$, a
fraction of milligram. This is a small scale in astrophysics and a large scale
in high-energy physics. It is reasonable to study the possibility that the
spectrum of the theory could include a stable or semi-stable non-perturbative
object at this scale: a Planck-mass quasi-particle. Recent developments in
classical general relativity (GR) and in LQG (LQG) add plausibility to this
possibility and suggest the existence of a semi-stable object at the mass

$m=\sqrt{\frac{\sqrt{3}\gamma c\hbar}{4G}}\sim 14\sqrt{\gamma}\;\mu g$ (1)

where $\gamma$ is the Barbero-Immizi parameter, a free parameter in LQG, akin
to the $\theta_{QCD}$ angle in quantum chromodynamics, and presumed of order
unity. These developments emerged from studying the dynamics of black holes.

We expect black holes to evolve into spacetime regions dominated by strong
quantum gravity effects. A number of recent lines of research have studied
these regions, and found an interesting scenario for the full evolution of a
black hole Rovelli2018h . Different ingredients contribute to this scenario.
These include a new solution of the Einstein equations Haggard2014  showing
that a trapping horizon can evolve into an anti-trapping one, a better
understanding of the interior of white holes and black holes Christodoulou2015
, and numerous applications of a variety of LQG techniques –canonical,
covariant, and numerical– to describe the genuinely non-perturbative regions
Modesto:2004xx ; Ashtekar:2005cj ; Rovelli2014ps ; Tavakoli2015 ; Bianchi2018e
; Lewandowski2022 ; Husain2022a ; Han:2023wxg ; Frisoni2023 ; Han2024 ;
Sobrinho2023 ; Borges2024 .

Three aspects of this scenario are particularly appealing. (a) It offers a
natural solution to the black hole information ‘paradox’. (b) It is in
principle, and perhaps even in practice, directly testable Perez2023 . And (c)
it provides a natural candidate for dark matter that does not require any new
physics (such as new fields, particles, or modifications of the field
equations): just general relativity and its quantum properties Rovelli2018f .
This scenario is compatible and possibly corroborating the general idea that
primordial black holes could play a key role for explaining dark matter, see
carr_primordial_2020 ; green_primordial_2021 .

This scenario combines distinct quantum phenomena happening in different
spacetime regions. It also includes dissipative as well as non-dissipative
aspects. The analysis of its various aspects employs different approximations
and truncations for treating different phenomena. Because of this complexity,
it can only be addressed ‘à la Fermi’, estimating the relevance and the import
of the various physical effects, rather than within a single mathematical-
physics idealization. This complexity motivates the present review paper,
which brings together several ingredients for this scenario, scattered in the
literature. The only truly original part of this review is Section III.6,
which present an argument supporting the idea that a Planck-mass quasi-
particle produced as a black hole remnant could be understood as a
superposition of a black and a white horizon.

We start with a quick sketch of the scenario (Section I.1) and an analysis of
the regions where classical GR is unreliable (Section I.2). Then we break the
presentation into two parts: a first part (Section II) where we discuss the
non-dissipative (time-reversal symmetric) aspects of the global dynamics of a
black hole, and a second part (Section III) where we take the dissipative
(time oriented) effects into account.

In the first, we discuss the bounce of a collapsing matter distribution
(Section II.1), namely the ‘Planck Star’ phenomenon, and the external metric
of the global black hole dynamics (Section II.2). A side section discusses the
counter intuitive classical geometry of white holes, because this play a role
in global picture (Section II.6). This part ends presenting the current state
of the calculations of the black-to-white transition probability
Christodoulou2018 .

In the second part, based on these calculations, we discuss the relevant
temporal regimes, the black hole information paradox (Section III.7), and the
structure of and stability of the remnants (Section III.5). The last part
(Section IV.1) focuses on the possibility of observations supporting the
scenario described. Unless indicated, we use Planck units $c=G=\hbar=k=1$.

###  I.1 A sketch of the scenario

Perhaps the most interesting discovery of the last decades is the abundance
and variety of the astrophysical object we now call black holes. Current
direct and indirect observations of these objects are all well accounted for
by classical GR. But classical GR is insufficient to account for the phenomena
that happen in their high-curvature regions, where genuine quantum gravity
effects are most likely non-negligible. Therefore GR does not provide a
reliable _global_ picture of the dynamics of these objects.

In a black hole produced by a collapsing matter distribution there are three
distinct spatiotemporal regions where we expect quantum gravity effects to
dominate: (i) the region when the matter distribution reaches Planck density;
(ii) the region, outside the matter distribution but inside the horizon, when
Planck curvature is reached; (iii) the region _outside_ the horizon, where the
Planck curvature is reached because of the shrinking of the horizon due to the
Hawking radiation. These events are spacelike with respect to one another.
Each region needs to be treated separately and understood on its own terms.
But all of them can be described using LQG as a guide, utilizing tools and
results from various branches of LQG, in particular the spinfoam amplitudes of
covariant LQG Rovelli2011c ; Rovelli2015 , Loop Quantum Cosmology
Ashtekar2015a ; agullo_loop_2016  and numerical calculations dona_how-
compute_2022 ; han_mathematica_2024 .

The intuition underpinning the scenario that emerges from combining these
results Bianchi2018e  is that quantum-gravitational pressure can stop
gravitational collapse and cause a bounce Rovelli2014ps , permitting the
entire content of the black hole to eventually dissipate Ashtekar:2005cj .
This is the same phenomenon as in loop quantum cosmology, that indicates that
the dominant quantum effect at high density is a quantum pressure sufficient
to counterbalance weight and reverse collapse.

This quantum gravitational pressure stops gravitational collapse when the
energy density becomes of Planckian order, yielding a new phase in the life of
gravitationally collapsed objects. At the bounce itself, the matter
distribution has maximal density and is called a “Planck star” Rovelli2014ps .
After this, the collapsing object bounces out. In the meanwhile, the geometry
outside the star, including the horizon, also bounces tunnelling Dona2024
from a black to a white hole geometry. This process can be short in proper
time, as well as, due to the huge gravitational time dilation, very long for
an external observer.

The process may seem incompatible with Birkhoff’s theorem (in the spherically
symmetric case), and this may be the reason for which the possibility for it
to happen was long been missed. As detailed later on, there is no
incompatibility, because the Birkhoff’s theorem is local but not global, hence
quantum tunnelling in a compact region can circumvent the theorem Haggard2014
and permit a black hole to evolve into an anti-trapped region, that is, a
white hole. This scenario has been explored in DeLorenzo2016 ;
Christodoulou2018 ; Bianchi2018e ; DAmbrosio2018 ; Rovelli2018a ; Fazzini2023
.

If the transition happens at the end of the evaporation, when quantum gravity
becomes relevant also outside the horizon, and where quantum gravitational
transition amplitudes indicate its probability to approach unity, most of the
energy of the black hole has already been radiated away into Hawking
radiation, which is a dissipative phenomenon that breaks time-reversal
invariance. Therefore, even if the transition itself can be described as an
elastic non-dissipative phenomenon, the overall life of a black hole is very
far from being time-reversal symmetric, and we may expect the white hole
emerging from the transition to be a remnant with mass at the Planck scale.

The internal geometry of these objects is a remarkably precise realization,
fully consistent with classical GR, of the ‘cornucopia’ intuition considered
in the 1980’s and later abandoned on the basis of arguments that (as we shall
discuss below) are not anymore cogent Banks1992b ; Banks1993 .

There is a well known instability affecting eternal and macroscopic white
holes: they can easily fall into a black hole as in the process described by
the classical Kruskal geometry. On the other hand, there are arguments
indicating that a Plank-mass white hole can be stabilized by quantum gravity
Rovelli2018f , by shifting it into a quantum superposition of black and white
geometries. As we show below, the key of this stabilization is the LQG area
gap, because of which there is likely no black (or white) hole with a mass
smaller than the Planck mass.

Hence the scenario predicts the existence of quasi-stable remnants of a known
mass, determined by the LQG area gap. A large number of these, of primordial
or possibly even pre-big-bang origin, would behave precisely as dark matter.
The possibility of direct detection of these object has been explored in
Perez2023  and we review it below.

Once quantum gravity phenomena are taken into account, the notion itself of
horizon is necessarily different than the one relevant for the classical
theory. _Event_ horizons looses their relevance, because they are likely not
to form at all. The Hawking radiation drags the dynamical evolution of the
horizon into regions where the Einstein equations do not hold anymore.
Accordingly, the discussion on the so called black-hole information paradox
needs to be reconsidered.

Even more dramatically, the traditional _definition_ of a black hole as an
object characterized by an _event_ horizon becomes misleading: in the light of
quantum gravity, the horizon of the realistic astrophysical black holes is
likely _not_ to be an _event_ horizon, because while the Kerr geometry and its
perturbations used in modelling astrophysical objects may be appropriate, its
future time evolution (which determine whether the horizon is an event
horizon) is not the one determined by the Einstein equations. Of course the
local properties of the horizon (described as apparent Hawking_Ellis ,
trapping Hayward1994 , isolated Ashtekar2000 , dynamical Ashtekar:2004cn , and
similar, see this last reference for a review) remain the same.

While the scenario illustrated takes fully into account quantum gravitational
phenomena in the high curvature region, it assumes that classical GR and
quantum field theory on curved spacetimes are good approximations outside
these regions. Contrary to what sometimes claimed, GR and quantum field theory
taken together, yield no inconsistencies in the entire region where quantum
gravity can be neglected. As argued in detail below, the common claim that
inconsistencies appear already at the Page time depends on strong versions of
the holographic hypothesis Rovelli2017a , or from confusion between ADM mass
and Bondi mass, between thermodynamic and von-Neumann entropies, or between
event and dynamical horizons. On the contrary, the scenario considered here is
only based on conventional GR and conventional quantum ideas.

Yet, this scenario implies a number or remarkable new phenomena: (i) Due to
the huge time dilation in a black hole, the process can last micro-seconds in
local proper time, but billions of light-years observed from the outside; (ii)
the internal volume of the hole can remain huge even as the size of the
horizon shrinks; (iii) the star’s bounce volume is much larger than Planckian,
because the onset of quantum-gravity effects is governed by density, not size;
(iv) the interior of an evaporating hole can keep memory of the initial state,
without information loss; (v) information can follow a different path from
energy, and be slowly released by the remnants, while most of the energy was
previously lost into the Hawking radiation. All this shows that unpalatable
phenomena like ‘firewalls’ Almheiri2013a  are not to be expected on the basis
of GR and quantum theory alone, unless one adopts arbitrary holographic
assumptions that are at best string-motivated only.

###  I.2 The domain of validity of classical gravity

Figure 1: The Carter-Penrose diagram of a Schwarzshild black hole until the
onset of quantum gravity. The light grey region is the collapsing star. The
dark grey region is where quantum gravity becomes relevant.

What happens inside a black hole is dynamical: it cannot be understood in
terms of any static or stationary model. The Carter-Penrose diagram of a
(classical) Schwarzschild black hole created from a gravitational collapse is
depicted on Figure 1. The dark grey region is where the classical theory
becomes unreliable, due to quantum gravitational effects. We expect this to
happen when the curvature becomes Planckian, for instance when the Kretschmann
scalar

$K^{2}=R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}=48\frac{M^{2}}{r^{6}}$
(2)

becomes of order 1 in natural units. Here $M$ is the black hole mass and $r$
is the Schwarzschild radius. This happens before the $r=0$ singularity inside
the black hole. Notice that if $M$ is macroscopic, it happens at a radius of
order $r\sim M^{\frac{1}{3}}$ much larger than the Planck radius ($r\sim 1$).

Importantly the surface where this happens is space-like.

Just outside the horizon, where $r\sim 2M$,
$K\sim\frac{M}{r^{3}}\sim\frac{1}{M^{2}}$. The Hawking evaporation steadily
decreases the mass $M$ of an isolated black hole, bringing $K$ up to Planckian
values, hence _the quantum region extends outside the horizon_. As we will see
later on, results in Christodoulou2018 ; Han2024  indicate that the transition
probability $P$ from black holes to white holes is proportional to

$P\sim e^{-M^{2}}.$ (3)

Thus becoming dominant at the end of the evaporation, where $M\sim 1$.

The possibility that quantum effects could appear earlier, at a (retarded)
time of order $M^{2}$ after the collapse, has been considered in Haggard2014 ;
Haggard20162  and in Husain2022 ; Husain2022a ; Fazzini2023  by considering
the interplay with the collapsing matter. The phenomenology this could give
rise to has been explored extensively in Barrau2014b ; Barrau2014 ;
Barrau2016c ; Vidotto2018 ; Bull2018 ; Barrau2018b . Here we will not consider
this hypothesis.

The region of the onset of quantum gravitational phenomena can thus be
organized into three sub regions Dambrosio2021  (see Figure 1 ):

  * •

Region a : The region which is neither directly causally connected to the
horizon nor to the collapsing matter distribution (it is _amid_ them).

  * •

Region b : The region in the vicinity of the horizon (_the boundary_).

  * •

Region c : The collapsing matter distribution region (_the center_).

The onset of the phenomena in these three regions are causally disconnected,
as they are separated by space-like distance. For a black hole of (initial)
mass $M$, this distance is of the order Dambrosio2021

$L\simeq{M}^{\frac{10}{3}},$ (4)

which is colossal for a macroscopic black hole. Notice that the locus of the
onset of quantum gravity is not ‘at a point’, but rather ‘at a time’. It is
not ‘somewhere’ but rather, ‘at some time’. Which in relativistic terms means
in a set of space-like related regions.

The fact that the various onsets of quantum gravity are spacially related
implies by locality that what happens in a region of this locus is causally
disconnected, hence independent from what happens in other regions. To imagine
that quantum gravity could violate this macroscopic causal disconnection is in
gross contradiction with what we know about reality, not supported or suggest
by anything we know, as far as we can see. To understand the physics of the
end of a black hole, we have to understand the quantum evolution of each of
these three regions independently.

##  II Non dissipative aspects of the transition

###  II.1 Planck stars

Let us start by studying what happens in the interior of a collapsing star,
after it enters its horizon, when it reaches Planckian density. The simplest
modeling of a collapsing matter distribution is provided by the Oppenheimer-
Snyder model Oppenheimer:1939ue : a spherically symmetric pressure-less
homogeneous matter distribution free-falling under its own weight. Assuming
the matter distribution distribution to start at rest in the distant past, the
metric inside a such a matter distribution distribution can be written in co-
moving and proper time coordinates $(T,R)$ as

$\mathop{}\\!\mathrm{d}s^{2}=-\mathop{}\\!\mathrm{d}T^{2}+a^{2}(T)(\mathop{}\\!\mathrm{d}R^{2}+R^{2}\mathop{}\\!\mathrm{d}\Omega^{2})\,,$
(5)

where the $T=constant$ slices define the homogeneity foliation,
$\mathop{}\\!\mathrm{d}\Omega^{2}$ is the metric of the unit 2-sphere,
$R\in[0,R_{boundary}]$ and $a(T)$ is known as the scale factor. The radial co-
moving coordinate of the boundary of the matter distribution distribution can
be chosen to be $R_{boundary}=1$ without loss of generality. The uniform
density of the matter distribution distribution is then $\rho=m/\frac{4}{3}\pi
a^{3}$, where $m$ is the total mass. Inserting this metric in the Einstein
field equations gives the Friedmann equation for $a(t)$:

$\frac{\dot{a}^{2}}{a^{2}}=\frac{8\pi}{3}\rho\,,$ (6)

where the overdot means differentiation with respect to $T$, which coincides
with the proper time. Eq. (6) can be solved:

$a(T)=\bigg{(}\frac{9m(T-T_{0})^{2}}{2R^{3}_{\text{star}}}\bigg{)}^{1/3}.$ (7)

Without loss of generality we can take the time at which the matter
distribution distribution collapses to zero physical radius to be $T=0$.

Figure 2: The scale factor $a(T)$ in eq. (10) that gives the standard LQC
bounce.

How does quantum gravity affects this dynamics? A major result in loop quantum
cosmology Ashtekar2006 ; Ashtekar:2006uz ; Agullo2013a ; Kelly:2020lec  is
that the Friedmann equation for the scale factor, Eq. (6), is modified by
quantum gravity effect into

$\frac{\dot{a}^{2}}{a^{2}}=\frac{8\pi}{3}\rho\,\Big{(}1-\frac{\rho}{\rho_{c}}\Big{)}\,,$
(8)

where $\rho_{c}=\sqrt{3}c^{2}/(32\pi^{2}\gamma^{3}\hbar G^{2})\sim c^{2}/\hbar
G^{2}$, $\gamma$ being the Barbero-Immirzi parameter, is the critical density
Ashtekar:2011ni , a constant with the dimension of a density and Planckian
value. This equation can be integrated to give

$a(T)=\bigg{(}\frac{9mT^{2}+Am}{2R^{3}_{\text{star}}}\bigg{)}^{1/3},$ (9)

where $A=3/(2\pi\rho_{c})$ is a parameter of Planckian value. In units where
$G=c=1$, the constant is $A\sim\hbar\sim m_{Pl}^{2}$. The last equation can be
integrated, giving

$a(T)=\left(\frac{9mT^{2}+Am}{2}\right)^{1/3}\,.$ (10)

Notice that $a(T)$ is positive for the whole range $T\in[-\infty,\infty]$: it
decreases for $T<0$, reaches a minimum $a_{0}=\sqrt[3]{Am/2}$ for $T=0$ and
then increases for $T>0$. See Fig. 2. This is the characteristic bounce of
loop quantum cosmology. This result shows that the line element in eq. (5) is
well defined everywhere, without singular collapse: the stars reaches a
maximal density, where it is called a ‘Planck star’, and then bounces and
expands.

Of course the time-reversal symmetric bounce is an approximation, because
dissipative effects break this symmetry. The Hawking radiation generates an
ingoing flux of negative energy that is likely to compensate the energy of the
star, in the interior of the hole. We will address these phenomena later on,
when discussing the dissipative aspects of the dynamics of the black hole.

To get a feeling of the physics of the bounce, we can rewrite (8) in the form

$\dot{a}^{2}=\frac{2m}{a}-\frac{Am^{2}}{a^{4}}.$ (11)

The coordinate $T$ is the proper time along the comoving worldlines, hence it
is also the proper time on the boundary of the star. This means that Eq. (11)
gives also the evolution of the physical radius $r_{b}(T)=a(T)$ of the matter
distributionin its own proper time, hence

$\dot{r}_{b}^{2}=\frac{2m}{r_{b}}-\frac{Am^{2}}{r_{b}^{4}}\,.$ (12)

This shows that a mass element on the boundary of the matter
distributionfalling in its own proper time feels a potential that is the
Newtonian one, precisely as in classical GR, but corrected by a repulsive
quantum term proportional to the inverse of the 4th power of the radius. This
is the short scale _repulsive_ force due to the quantum pressure, that grows
stronger at smaller radius.

Is this dynamics compatible with the dynamics of the surrounding geometry?

###  II.2 Black-to-white transition

In the classical Oppenheimer-Snyder model, the geometry of the collapsing
matteris compatible with a surrounding Schwarzschild geometry. That is,
matching conditions between the two geometries are satisfied on the boundary
of the matter distribution. To confirm this, consider the surface bounding the
collapsing matter as a free falling spherically symmetric shell in a
Schwarzschild metric

$\mathop{}\\!\mathrm{d}s^{2}=-F(r)dt^{2}+dr^{2}/F(r)+r^{2}\mathop{}\\!\mathrm{d}\Omega^{2}\,,$
(13)

where $F(r)=1-2m/r$. The shell follows a radial time-like geodetic and
therefore satisfies

$-1=-F(r)\dot{t}^{2}+\dot{r}^{2}/F(r)$ (14)

The conservation law associated to the time-like Killing symmetry gives
$F(r)\dot{t}=E$ where $E$ is a constant that can be taken equal to unit if the
shell starts at rest at infinite radius. Hence $-F(r)=-E^{2}+\dot{r}^{2}$,
that is

$\dot{r}^{2}=\frac{2m}{r}.$ (15)

A comparison with (12) shows that this is exactly the change of the
Schwarzschild radius in proper time of the boundary of the star, in the
classical ($A$=0) case.

But this suggest immediately the form of a metric which is compatible with the
bouncing star, in the quantum case: it is again (13) but with

$F(r)=1-\frac{2m}{r}+\frac{Am}{r^{4}}.$ (16)

This is an interesting metric. (For earlier attempts to write the black hole
metric in the quantum region see for instance Modesto2004 ; Hossenfelder2010a
; Gambini2014a ; Olmedo:2017lvt ; Malafarina2017 ; Ashtekar2018 ; Clements2019
; Bodendorfer2019a ; Volovik2021 ; BenAchour2020 ; Munch2020 ; Kelly:2020lec )
It was suggested already in Rovelli2014ps  and has been derived by various
quantum gravity research groups, using different methods Kelly:2020uwj ;
Giesel2022 ; Munch2020 ; BenAchour2020 ; Lewandowski2022 ; Bobula2023 ;
Fazzini2023 , as a credible candidate for the effective metric in the high
curvature of a spherically symmetric black hole. For instance, it is uniquely
determined by requiring it to satisfy matching conditions with the collapsing
matter distribution and keep the Killing symmetry of the Schwarzschild
geometry (which in the interior of the horizon is space-like, not time-like as
in the exterior.)

This metric has inner and outer Killing horizons. See Han:2023wxg  for full
details. For $m\gg m_{Pl}$, that is $m^{2}\gg A$, the outer one is at

$r_{+}=2m+O(A/m)\sim r_{Schwarzschild}$ (17)

in the classical region. If $m$ is large, this is a negligible modification of
the usual Schwarzschild horizon. While the inner one is at

$r_{-}=\sqrt[3]{Am/2}+O(A^{2/3}/m^{1/3})\sim\sqrt[3]{m/m_{Pl}}\,l_{Pl}$ (18)

deep inside the quantum region, where the spacetime curvature has Planckian
size. These are all also apparent horizons: they separate trapped, non-trapped
and anti-trapped regions Han:2023wxg .

Figure 3: Conformal diagram of the maximal extension of the spacetime
representing the matter distribution and the exterior region defined by eqs.
(13) and (16).

The Carter-Penrose diagram of the maximal extension of this metric is depicted
in Figure 3, which shows the relative location of these horizons.

Interestingly this geometry has a global structure (including the inner
horizons) similar to the Kerr, the Reissner-Nordström, and the general Kerr-
Newman black holes. (The full bounce in the Reissner-Nordström case has been
described in Rignon-Bret2021 , while the Kerr case has not been constructed
yet.)

The maximal extension of this geometry has (i) two asymptotic regions and (ii)
a time-like singularity in a high curvature region. These two features make it
an unlikely candidate for describing the actual physics of realistic black
holes. But a surprising result on spherically symmetric solutions of the
Einstein’s field equations, solve both these difficulties. This is illustrated
in the next section.

###  II.3 The exterior metric

The Birkhoff’s theorem is sometimes presented as stating that the only
spherically symmetric spacetime compatible with the vacuum Einstein’s field
equations is the Kruskal geometry. At a sufficient distance from a black hole,
we expect quantum effects to be negligible and therefore the theorem to hold.
Hence we might expected that whatever happens where quantum effects are
negligible must form a subset of a Kruskal geometry. In the Kruskal metric,
there is an anti-trapped region, but it is before, not after the trapped
region. Hence there seem no way to join the metric described above to a single
external asymptotic region. But the above is wrong. The reason is that
Birkhoff’s theorem is local, not global: it states that a spherically
symmetric solution of the vacuum Einstein field equations is _locally_ , and
not necessarily globally, isometric to the Kruskal metric. This seems a red
hair, but has momentous consequences for understanding the dynamics of quantum
black holes.

In fact, as surprisingly realized in Haggard2014 , there is an _exact_
solution of the Einstein’s field equations, with a single asymptotic region,
that can surround a quantum transition from a black to a white hole. The
Carter-Penrose diagram of this solution is depicted in the left hand side of
Figure 4, while the right hand side shows how this can be _locally_
isometrically mapped onto the Kruskal spacetime. The point is that the map is
not injective, hence the (pink) classical part of the spacetime is locally
isomorphic but not a subset of the Kruskal geometry.

Figure 4: The spacetime discovered in Haggard2014 , describing the collapse of
a null shell into a black hole and its bounce out a white hole (left); and the
way it can be locally isometrically mapped onto the Kruskal spacetime (right).
The pink region is an exact solution of the classical Einstein equations.

This solution describes a null shell collapsing into a black hole and bouncing
out from a white hole. The key point is that the black hole (the trapped
region) is in the past of the white hole (the anti-trapped region) while in
the Kruskal spacetime it is the opposite. This shows that an exact solution of
the Einstein equations is compatible with a quantum process happening in a
compact high curvature region tunnelling a black into a white hole.

This same idea solves the two difficulties mentioned at the end of last
section, providing a spacetime with a single asymptotic region and no
singularities for the bouncing star. This can be constructed by cutting and
gluing the maximal extension of the metric defined by eqs. (13) and (16), as
follows.

First, pick a point $\alpha$ in the interior region, on the surface invariant
under time reversal and a point $\beta_{L}$ outside the horizons, in the first
asymptotic region, as in the left panel of Figure 5 This choice depends on
three parameters: the advanced times $v(\alpha)$ and $v(\beta_{L})$ and the
Schwarzschild radius of $\beta_{L}$. Next, consider the blue line of the
figure. This has a null portion joining $\alpha$ and $\beta_{L}$ with their
last common past event and a spacelike portion joining $\beta_{L}$ to
spacelike infinity along a constant Schwarzschild time. Next, draw the line
symmetric to this under time reversal, as in the right panel of Figure 5.

Figure 5: The construction of the single asymptotic region spacetime, I:
choosing the cut and glue surface

Next, delete the entire spacetime region enclosed within the blue line, and
glue its two space-like portions as in Figure 6. This is clearly possible
since they are both constant Schwarzschild time surfaces.

Figure 6: The construction of the single asymptotic region spacetime, II:
deleting the singular region and gluing the two equal time exterior surfaces.

The resulting spacetime has a single asymptotic region, is everywhere locally
isomorphic to the maximal extension of the metric (13-16), and has a hole,
depicted as the $B$ diamond of Figure 6. In Han:2023wxg  it is shown that the
$B$ region admits a regular Lorentzian metric compatible with rest of the
spacetime geometry. The $B$ region and its physics will be discussed later on.
For the moment, let us focus on the rest of the spacetime.

The popular idea that at the end of the evaporation a black hole disappears is
not supported by any theory and contradicts unitarity. The above construction
offers a far more plausible alternative. See Figure 7.

Figure 7: Left: a popular idea of what happens at the end of the evaporation.
This is not supported by any theory and contradicts unitarity. On the right,
the plausible scenario.

The relation of the resulting geometry with the various quantum gravity
phenomena mentioned above is depicted in Figure 8. (See also Figure 8 in
Carballo-Rubio2021 .)

Figure 8: The Carter-Penrose diagram of the black-to-white transition. The
dark grey region is the quantum gravity region. The black hole (trapped
region) is below the quantum gravity region while the white hole is above. The
trapping horizons are the dashed lines.

We have constructed the above metric introducing three parameters
($v(\alpha),v(\beta_{L}))$ and the Schwarzschild radius of $\beta_{L}$. The
resulting geometry (a part from the choice of the metric inside $B$) depends
then on these parameters, plus the mass of the star. Of the three parameters,
two simply locate the $B$ region: they determine the minimal and maximal
radius of the diamond defining it. The last one is far more interesting. It
determines the global geometry of the spacetime, in the sense in which the
radius of a cylinder determines the global geometry of a locally flat
cylinder. Its geometrical and physical interpretation can be seen as follows:
it determines the time $T$ at which an observer at large distance from the
hole sees the duration of the process.

Figure 9: The definition of the duration of the process. The red line
represents an observer at fixed radius.

This is illustrated in Figure 9, where the red line represents an observer at
a fixed radial distance $R$. As shown in Han:2023wxg , this duration is

$T=2R+4m\ln{r-2m}-4m\ln\delta.$ (19)

The first term on the right hand side can be interpreted as the back and forth
travel time for the light from the observer to the hole. The second term is
the standard relativistic correction. The last term is independent from the
radius and represents a property of the transition itself. The quantity
$\delta$ can be measured by distant observers by measuring $T$ at different
values of $R$. A little geometry shows that it depends on the choice of
$\beta_{L}$ as follows: $r=2m+\delta$ is the radius at which the constant
Schwarzschild-time surface passing by $\beta_{L}$ intersects the surface of
the falling star. This is clearly very close to the collapse (the point where
the surface enters entirely in its horizon), if the lifetime of the hole is
long. Thus

$\tau=-4m\ln\delta$ (20)

can be taken as a definition if the duration of the quantum process, as
observed from infinity. In our construction, it is determined by how the
geometry has been constructed cutting and gluing. Operationally, it is the
observed duration of the process from very large distance. Physically, it is
determined by the quantum theory of the geometry of the $B$ region, to which
we now turn.

###  II.4 The Boundary Region

In the previous sections, we have constructed a spacetime geometry that could
describe the evolution of a black hole past the region where quantum
gravitational effects dominate. How well supported by known physics is this
picture?

The bounce of the matter distribution is supported by the LQG modelling of the
quantum dynamics of symmetric spacetimes. The physical plausibility of the
effective metric outside the matter distribution but inside the outer horizons
is also supported by various LQG modelling of the quantum dynamics of
symmetric spacetimes. The structure of the $B$ region, on the other hand, has
only been guessed as a plausible solution to the requirement of a global
dynamics compatible with the external geometry dictated by the classical
Einstein equations. Does it follow from quantum gravity? Is it allowed by it?

To address this question we need to step back and think more precisely what we
are doing. First, let us remind once more that for the moment we are
disregarding the dissipative effects that break time reversal invariance.
These will be studied in the next section. Then, consider the fact that we are
describing a quantum process. There are different ways of describing quantum
processes. In some regimes, these can be described as corrections to the
classical equations of motion. This is not viable for the quantum transition
of the horizon, which is a non perturbative process. One cannot describe
quantum tunnelling as a simple $\hbar$ corrections to classical trajectories.
Tunnelling is non analytical in $\hbar$.

Another possibility is to describe the evolution of a quantum state. Quantum
states that approximate classical configurations, such as coherent states,
spread. We could therefore consider the full quantum state of the geometry
after the actual quantum process as a quantum states, namely a quantum
superposition of geometries. This is analogous to describing nuclear
radioactivity in terms of a wave function of the emitted particle widely
spread in space and time. Correct, but incomplete. What we observe in nuclear
radioactivity is not the wave function of the emitted particle, widely spread
in space and time. Rather, it is a specific space and time location where the
particle is detected, say by a Geiger counter.

In Copenhagen terms, we observe the result of a measurement. In Many World
terms, we observe the position of the particle in one branch, after
decoherence. In Relational Quantum Mechanics, we observe the metric relative
to us, again after decoherence. The theory, of course, does not predict any
specific outcome, but it does predict the probability of the alternatives.

To compute these, we need to compute transition amplitudes. This is how we can
address the problem of describing the $B$ region in LQG.

That is, we can compute the transition amplitude for the geometry to make a
quantum transition from its value in the past boundary of $B$ to its future
boundary (as opposite to not yet making it, as in radioactivity). Notice that
the geometry (both intrinsic and extrinsic) of these two boundaries is known
from the construction of the above section. We have simply to compute the LQG
transition amplitudes between quantum states approximating these geometries.

The covariant formulation of LQG is precisely formulated to do this
calculation, using approximations provided by truncations of the degrees of
freedom. The calculation has been performed, under some drastic
simplifications by Christodoulou and D’Ambrosio using analytical techniques in
Christodoulou2018 ; Dambrosio2021 ; Christodoulou2023 . The result has been
confirmed numerically in Frisoni2023  and Han2024 . Numerical calculations are
currently developed to go beyond these approximations. The way these
calculations are performed is summarized in the next section.

###  II.5 The LQG transition amplitude

The covariant formulation of LQG defines transition amplitudes between quantum
states of the geometry, in suitable truncations of the numbers of degrees of
freedom. The number of degrees of freedom can be truncated by approximating
the 3d geometry of the boundary by means of a cellular decomposition. The two
skeleton of the dual of the decomposition form a graph and the truncated
variables can be taken to be $SU(2)$ group elements $U_{l}$ on the links $l$
of this graph. The quantum states of the geometry can then be taken to be
functions of the $U_{l}$’s in the space $L_{2}[SU(2)^{L}/SU(2)^{N}]$, where
$L$ and $N$ are respectively the numbers of links and nodes of $\Gamma$, as in
lattice Yang Mills theory. The algebra formed by the group elements $U_{l}$
themselves and the Left invariant operators is the observable algebra, which
has a natural interpretation in terms of functions of the discretized
geometry. A large literature has developed a theory of coherent states for
this algebra, representing semiclassical geometrical states.

Transition amplitudes can be equally understood as Feynman sums over
geometries, or as analogous to lattice Yang-Mills calculations. Concretely,
they are defined in a truncation. A truncation is here given by a two-complex
bounded by the boundary graph. The amplitude is defined by the product of the
vertex amplitudes

$A_{v}(\psi)=P_{SL(2,C)}Y_{\gamma}\psi(1\\!\\!1),$ (21)

where $P_{SL(2,C)}$ is the projector on the $SL(2,C)$ invariant part and
$Y_{\gamma}$ is the ‘simplicity map’ from $SU(2)$ to $SL(2,C)$ representations
defined, in the canonical basis, by

$|m;j\rangle\mapsto|\gamma j,j;j,m\rangle.$ (22)

See Rovelli2011c ; Rovelli2015  for the full details. This amplitude has been
shown to give the Einstein dynamics in suitable limits Barrett2009  and can be
taken as a definition of the covariant LQG theory.

To apply this general theory to the $B$ region we need to find a suitably
simple cellular decomposition of this region. Steps in this direction have
been taken in Soltani2021 , and a numerical calculation has been achieved in
Han2024 . An analytical calculation, on the other hand, has been completed
only in a simpler setting, where the entire quantum region is discretized.

Figure 10: The boundary of the quantum region for the simplified calculation
of the transition amplitude.

The complexity of the calculation on the $B$ region alone is given by the
topology of $B$, which is the product of a two-sphere and a disk. The disk is
the product of a finite time interval and a finite radial interval and it is
delimited by an exterior two-sphere $S_{+}$ that surrounds the horizon and by
an interior two-sphere $S_{-}$ surrounded by the horizon (sitting on the
bounce radius of the transition of the internal geometry of the black hole).
The two two-spheres $S_{+}$ and $S_{-}$ split the boundary $\Sigma$ of $B$
into a past component $\Sigma^{p}$ and a future component $\Sigma^{f}$.

Considering instead the entire quantum region simplifies the topology. The
boundary of the quantum region can be take to be as depicted in blue Figure
10. The tip, at the largest radius, is a two sphere, chosen outside the
Planckian curvature region. The lowest (past) part of the boundary and the
upper (future) one are both 3d balls. Each of this can be triangulated with 4
tetrahedra and the interior can be triangulated with two four-simplices
sharing an internal tetrahedron, as depicted in Figure 11.

Figure 11: The graph of the spin network and the two complex of the spinfoam
for the calculation of the tranisition.

We refer to Christodoulou2018 ; Dambrosio2021 ; Christodoulou2023  for the
analytical calculation.

The numerical calculations have been performed using different techniques.
Frisoni2023  has analised the simpler geometry, utilizing the sl2cfoam-next
code for computing spinfoam transition amplitudes. A good introduction to
this, and full references, is in Dona2022a . The code permits to explore the
low-spin (small-mass) regime of the amplitude. On the other hand, Han2024
utilizes the complex saddle point method for computing spinfoam amplitudes,
which permits to explore the high-spin (large-mass) regime Han2020 ; Han2020 ;
Han2023 ; Han2024a . A common result of all these investigations (that
contradicts some previous expectations) is that the transition amplitude gives
a transition probability proportional to

$P\sim e^{-\frac{Gm^{2}}{c\hbar}}\sim e^{-(\frac{m}{m_{P}})^{2}}.$ (23)

Notice that this is indeed non-analytic in the Planck constant. Therefore it
is a genuine non perturbative result that cannot be obtained in conventional
perturbation theory expanding around a classical solution. The exponential is
characteristic of quantum tunnelling phenomena. The transition, in fact, is
forbidden classically (classically the geometry evolves into a singularity)
and is characteristic quantum tunnelling effect.

Another way of understanding this result is to connect to the asymptotic
behaviour of the amplitudes

$W\sim e^{iS_{Regge}}\sim e^{i\sum_{f}j_{f}\Theta_{f}(j)},$ (24)

where $S_{Regge}$ is the Regge action: a sum over the faces of the face spin
times the dihedral angle of the face, which is a function of all the spins,
determined by the interpolating flat geometry. Here there is no interpolating
geometry and the angles turn out to be imaginary, giving

$W\sim e^{-c\sum_{f}j_{f}}\sim e^{-c\sum_{f}A_{f}},\sim e^{-c\;m^{2}}.$ (25)

The simplest way to interpret this result is in analogy to nuclear
radioactivity: as a transition probability per unit of (here Planck-) time.
The immediate consequence of this result is that the transition probability is
exponentially suppressed if the mass of the hole is larger than the Planck
mass.

Anticipating the discussion about dissipative effects, we can see that toward
the end of the Hawking radiation, $m$ decreases and approaches $m_{P}$. At
this stage the transition becomes increasingly probable, going to probability
unit when the mass becomes actually Planckian.

In other words, the calculation indicates that the transition can happen and
it is likely to happen, but only at the end of the Hawking evaporation, giving
birth to a white hole with near Planckian mass.

###  II.6 White holes

Before addressing the dissipative aspects of the life of a hole, let us pause
to clarify a few properties of the white holes. White holes, like black holes,
are exact solutions of the Einstein’s equation. Precisely like black holes,
they have long been expected not to play any role in our universe, by a
majority of physicists. The situation has changed for black holes in the last
decades. It has not yet equally changed for white holes, but it might soon.

A white hole spacetime is simply the time reversal of a black hole spacetime.
For instance, a classical black hole formed by a collapsing matter
distribution and its time reversal, a white hole from which matter
distribution emerges, are depicted in Figure 12.

Figure 12: The spacetimes of a black and a white hole (outside the respective
stars, as subsets of the Kruskal geometry).

The reason black holes have been traditionally taken more seriously as
candidates for real objects than white holes is that we have a well understood
classical scenario for how a black hole can form, but not so for a white hole.
Since the two are the time reversal of one another, this implies that we have
a well understood classical scenario for how a white hole can end, but not so
for a black hole. The end of a black hole is definitely a quantum phenomenon,
because it involves high curvature regions, where quantum theory cannot be
neglected. The same must be true for the birth of a white hole.

All this points to a very simple possibility: the end of a black hole is the
birth of a white hole, via a quantum transition process. This is precisely the
scenario we are studying here. A white hole can be originated by a dying black
hole.

The difference between a black hole and a white hole is not very pronounced.
In fact, observed from the outside (say from the exterior of a sphere of
radius $r=2m+\epsilon>2m$, where $m$ is the mass of the hole) and for a finite
amount of time, a white hole cannot be distinguished from a black hole.

This is clear from the usual Schwarzschild line element, which is symmetric
under time reversal, and therefore describes equally well the exterior of a
black hole and the exterior of a white hole. Equivalently, zone II of the
maximal extension of the Schwarzschild solution is equally the outside of a
black hole and the outside of a white hole (see Fig. 1, Left). Analogous
considerations hold for the Kerr solution. The continuation of the external
metric of a stationary Kerr or Schwarzschild spacetime inside the radius
$r=2m+\epsilon$ contains both a trapped region (a black hole) ad an anti-
trapped region (a white hole).

Figure 13: Left: in the extended Schwarzschild spacetime, which is stationary,
the (light grey) region outside $r=2m+\epsilon$ (dotted line) is equally the
outside of a black and a white hole. Center: A collapsing matter distribution
(dark grey) replaces the white hole region (“WH”) in the non-stationary
collapse metric. Right: The time revered process. The difference between the
last two can only be detected looking a the past, or the future.

What distinguishes then the objects we call ‘black holes’ from ‘white holes’?
The objects in the sky we call ‘black holes’ are described by a stationary
metric only approximately and for a limited time. We expect that (at least) in
the past their metric was definitely non-stationary, and they were produced by
gravitational collapse. The energy contained inside $r=2m+\epsilon$ was less
than $m$ in the past and the continuation of the metric inside this radius
contains the trapped region, but not the anti-trapped region, which is instead
replaced by the region describing the collapsing star. Therefore seen from the
outside a ‘black hole’ (as opposite to a ‘white hole’) is only characterised
by the fact that in the past it does not have an anti-trapped region (see Fig.
1, Center). Viceversa, a white hole is an object that from the exterior and
for a finite time is indistinguishable from a black hole, but in the future
ceases to be stationary, the amount of energy inside $r=2m+\epsilon$
decreases, and there is no trapped region in the future (see Fig.1, Right).
All this regards classical gravity only.

##  III Dissipative aspects of the transition

###  III.1 Black hole lifetime

The most relevant dissipative phenomenon in the life of a black hole is the
Hawking radiation. This is a markedly irreversible process that dissipates the
energy of the collapsed matter distribution into the heat of the emitted
thermal radiation.

The lifetime $\tau_{BH}$ of a black hole is known by Hawking radiation theory.
It can be estimated as follows. The Hawking radiation is thermal, namely it
has Planck spectrum. The peak of the spectrum is on a wavelength $\lambda$
that is determined by the only length parameter in the problem: the
Schwarzschild radius $2Gm/c^{2}$. Therefore the radiation is mostly composed
by quanta of frequency

$\nu=c/\lambda\sim\frac{c^{3}}{Gm}$ (26)

These have an energy $E=h\nu$ and therefore (being thermal) a temperature $T$
given by

$kT=E\sim\frac{c^{3}\hbar}{Gm}$ (27)

This is (up to a numerical factor) Hawking’s temperature. The emission of the
horizon can be modelled as the emission of a sphere at this temperature with
the area of the horizon. This gives an emitted power (going back to natural
units)

$P=\frac{dm}{dt}\sim AT\sim m^{2}m^{-4}=m^{-2}.$ (28)

This is a differential equation for $m(t)$ that can be immediately integrated
giving $m^{3}\sim t$. Therefore we expect that the finite lifetime of a black
hole due to the Hawking evaporation is of the order

$\tau_{BH}\sim m_{o}^{3}$ (29)

where $m_{o}$ is the _initial_ mass of the hole.111The effect of the back
reaction on the metric of the full bounce has been studied in
martin2019evaporating . See references there.

The evaporation shrinks the horizon, bringing it close to Planckian size,
where the black-to-white transition has high probability to happen. The white
hole generated by the process has then an horizon of Planckian size. (This
lifetime can be shorter if quantum gravity fluctuations trigger an earlier
tunnelling. This could be as early as $\tau_{BH}\sim m_{o}^{2}$ as suggested
in Haggard2014 ; Haggard20162 . Here we only focus on the possibility that the
transition happens at the end of the evaporation.)

However, the fact that the black hole has evaporated does not mean that it is
“small” in any sense of the word. In fact, its interior is vast. To understand
this point, a detour on the size of the interior of black holes is important,
as this is a frequently misunderstood and misleading issue.

###  III.2 How big is the interior of a black hole?

In Minkowski spacetime, we say that a space-like 2-sphere $S$ of radius $r$
encloses a space-like ball of volume $\frac{4}{3}\pi r^{3}$. As a 3d surface
in Minkowski space, this ball is characterized by two properties: it is
linear, and it is the surfaces that maximizes the volume among those bounded
by $S$. Any deformation, in fact, is time-like and _decreases_ the volume.

In a curved Lorentzian manifold, linearity is meaningless, but we can still
talk about the volume enclosed in a spacelike (topological) 2-sphere $S$ by
defining it as the volume of the 3d surface that _maximizes_ the volume, among
all 3d surfaces bounded by $S$. This definition, introduced in
Christodoulou2015 , allows us to talk of the interior volume of a black hole
in a rigorous manner, and provides a simple intuition of the internal geometry
of a black hole by defining a natural foliation of this geometry.

Consider therefore the spherical symmetric spacetime of a collapsing matter
distribution and choose a retarded time $v$. The intersection of the retarded
time with the horizon defines a 2-sphere $S_{v}$. Let $V(v)$ be the volume of
the maximal-volume spacelike ball bounded by $S_{v}$. This is the volume of
the interior of the black hole at the retarded time $v$ on its horizon. The
dependence of this volume on $v$ has been computed both for an eternal horizon
and for an evaporating one Christodoulou2015 ; DeLorenzo2016 ; Bengtsson2015 ;
Ong2015 ; Wang2017 ; Christodoulou2016a : it grows linearly in $v$.

Specifically Christodoulou2015 ; Christodoulou2016a  that, at advanced time
$v$ after the collapse, a black hole with mass $m$ (disregarding evaporation)
has interior volume

$V\sim 3\sqrt{3}\pi\;m^{2}v$ (30)

for $v\gg m$.

Recall that the interior of the horizon is never stationary: it is dynamical.
(The common expression “stationary black hole” means a black hole whose
_external_ geometry is stationary.) In the natural foliation considered above,
the internal volume of the black hole increases steadily in time. More
specifically, the interior is like a tube whose radial dimensions shrink in
time while its longitudinal dimension increases, in such a way that the total
volume increases.

This implies that an old evaporated black hole has a small horizon _but a huge
internal volume_. Ignoring this fact has nourished a wrong intuition about the
geometry of evaporated black holes. Contrary to what is sometimes believed,
these are genuinely different from young black holes with the same mass.

In other words, we must not be fooled by the no-hair theorem. This theorem
states that two black holes that have the same mass, charge and angular
momentum, classically settle on the same _external_ geometry, but certainly
does not say that they have the same interior! Older ones are bigger. In fact
they can have vastly different interiors. Accordingly, in the quantum theory,
the full quantum state of a black hole will certainly not be determined by
external quantities like mass, charges and angular momentum only: there will
be quantum numbers $v_{i}$ distinguishing holes with the same exterior but
(possibly vastly) different interiors.

At the moment of the quantum transition therefore an old (previously)
macroscopic black hole is an extremely long geometrical entity. It is radially
shrinking, but growing in length. The transition is a bounce that reverses the
process: the interior of a white hole is expanding radially, and shortening.
See Figure 14.

Figure 14: A sketch of evolution of the internal geometry of the hole at the
bounce (up to some quantum gravity phenomena, see later).

###  III.3 Remnants and their lifetime

The argument of the previous section suggest that at the end of the
evaporation a black hole undergoes a quantum transition to a white hole with a
Planckian-size horizon and a vast interior.

The possibility of remnants with a similar structure was considered in the
1990’s Giddings1992b . What was not realized at that time is that classical GR
does in fact predict the existence of objects with the these properties: white
holes; and quantum theory could account for their formation at the end of the
evaporation and (as we shall see below) for their stability.

Before studying more in detail the quantum structure of these remnants, let us
ask what could be their lifetime.

A number of arguments suggest that this should be long. First, the Einstein
equations state that it takes a long (advanced) time to generate a black hole
with a large interior. Time reversing, they also state that it takes a long
(retarded) time to dissipate a white hole with a large interior.

The second argument concerns the information trapped inside the hole. The
Hawking radiation is genuinely thermal: it is describe by a quantum state
which is not pure: it is a density matrix. This is because the Hawking
radiation that escapes to infinity forms together with negative radiation that
falls inside the hole. The two are entangled, therefore the part escaping to
infinity has von Newman entropy. The total entropy of the Hawking radiation is
of the order of the Bekenstein Hawking entropy

$S_{BG}=\frac{A}{4}.$ (31)

where $A\sim m_{o}^{2}$ is the _initial_ area of the black hole, before the
beginning of the evaporation. Some scientists expect that the total entropy of
the Hawking radiation start decreasing at Page time because late Hawking
quanta are correlated with early ones. We are not convinced by this
expectation for reasons that we explain in detail later on. Hence an amount of
information of the order $S_{BH}$ is trapped inside the hole during the
quantum transition. This must be emitted later in the form or radiation by the
remnant. The remnant has total energy of the order of the Planck mass, namely
of order unit in natural units. It must therefore be able to emit a large
amount of information by emitting very small energy.

To emit a lot of information within small energy, we need a large number of
very low-energy quanta. These must have very low frequency, and hence require
a long time. As shown in Preskill1993 ; Marolf2017a ; Kazemian2022a  (and with
an argument detailed in Section IV.5) this leads to a lower bound in emission
time:

$\tau_{W\\!H}\sim m_{o}^{4}.$ (32)

Here $m_{o}$ is the mass of the hole before the evaporation (which determines
the amount of information in the hole).

Notice that the state of the black hole at some given (advanced) time, which
determines its full future evolution, is not specified by its sole current
mass $m$, but also by the internal geometry, which in turn is determined by
the initial mass of the black hole.

We can account for this by writing the quantum state of the black hole at some
given time in the form $|m_{o},m\rangle_{B}$, where the first quantum number
is the initial mass, which determines the size of the interior, while the
second is the current mass which determines the area $A=16\pi m^{2}$ of the
horizon on the given time slice and decreases in the evaporation. We do not
need to keep track of other possible quantum numbers here.

At formation, the hole is in the state $|m_{o},m_{o}\rangle_{B}$, then $m$
decreases by Hawking evaporation until the state
$|m_{o},m_{P\ell}\rangle_{B}$. This states tunnels to a white hole state with
the same quantum numbers, which we denote $|m_{o},m_{P\ell}\rangle_{W}$. The
tunnelling process itself from black to white is short and takes a time of the
order of the current mass Christodoulou2018 ; Barcelo2016 . Here is therefore
a first account (which we correct below) of the full life cycle of a
gravitationaly collapsed object

$\displaystyle\xrightarrow[\text{collapse}]{}|m_{o},m_{o}\rangle_{B}\xrightarrow[\text{black
hole}]{\tau_{W\\!H}\sim
m_{o}^{3}}|m_{o},m_{P\ell}\rangle_{B}\xrightarrow[\text{tunnelling}]{\tau_{T}\sim
m_{P\ell}}|m_{o},m_{P\ell}\rangle_{W}\xrightarrow[\text{white
hole}]{\tau_{W\\!H}\sim
m_{o}^{4}}|m_{P\ell},m_{P\ell}\rangle_{W}\xrightarrow[\text{full
dissipation}]{}.$ (33)

These are time scales for a distant observer. If $m_{o}$ is large, the process
is very long in the time of a distant observer. But it is extremely short (of
order $m_{o}$, which is the time light takes to cross the radius of the star)
if measured on the bouncing matter distribution itself. The huge difference is
due to the extreme gravitational time dilation Haggard2014 . Time slows down
near high density mass. An observer (capable of resisting the tidal forces)
landing on a Planck matter distribution will find herself nearly immediately
in the distant future, at the time where the black hole ends its evaporation.
The _proper_ lifetime of a Planck matter distribution is short: from its own
perspective, the matter distribution is essentially a bounce. A black hole is
a shortcut to the distant future.

This is a compelling scenario, but it still need to be corrected.

###  III.4 Instability

A classic macroscopic white hole is an unstable solution of the Einstein
equations (see Chapter 15 in Frolov2012  and references therein). This means
that there are solutions with initial data that are arbitrarily close to the
white hole initial data, but have a qualitatively different future.

The qualitative different future is the formation of a black hole in the
future of the white hole. That is, a white hole is unstable toward becoming a
black hole. The transformation of a white hole into a black hole is similar to
the process described by the vacuum Kruskal spacetime.

To see why there is this instability it is easier to address the time reversed
scenario first, because we have a better intuition about it: to show that
solutions with _final_ data arbitrarily close to those of a _black_ hole, must
have a qualitatively different past from the black hole. Or in other words,
that the data on future null infinity on a black hole spacetime cannot admit
certain arbitrary small variations.

To see this, consider a black hole of mass $m$, formed by a collapsed star. In
the unperturbed solution, no energy reaches future null infinity (here we are
considering classical physics: no Hawking radiation). Now, consider a
perturbation of this spacetime in which there is a small spherically symmetric
pulse of null radiation with total energy $\epsilon$ arriving on null
infinity. Here $\epsilon$ can be arbitrarily small. In order to arrive to null
infinity, this radiation must be emitted by the surface of the matter
distribution before this enters the horizon. Say it was emitted when the
radius of the matter distribution was $2m+\delta$. But this must also be
outside the Schwarzschild radius of the matter distribution plus the energy to
be emitted, otherwise it could not reach null infinity either. Hence
necessarily $2m+\delta>2m+\epsilon$ that is, we must have $\delta>\epsilon$.
Now let’s $u_{\delta}$ the retarded time of the emission point. Necessarily,
the energy pulse of energy $\epsilon$ must reach null infinity _before_
$u_{\delta}$. Therefore, no perturbation with energy $\epsilon$ can reach null
infinity _after_ $u_{\delta}$. That is, we cannot perturb the the final data
arbitrarily. However small is $\epsilon$ there are locations where it cannot
get to.

Can a perturbation with energy $\epsilon$ reach null infinity _after_
$u_{\delta}$, with some different past? Yes it does! A white hole of mass
$2m+\epsilon$ can emit the pulse and then become a black hole of mass $m$! In
this case, the pulse is never at a radius where it can trigger an increase of
the size of the blackhole.

Time reversing this scenario shows immediately why a classical eternal white
hole of mass $m$ is unstable: any arbitrary small pulse originating from past
null infinity sufficiently early in time will bring enough energy to be inside
the Schwarzschild radius before reaching the matter distribution emerging from
the white hole, thus triggering the formation of a black hole.

In other words, the spacetime depicted in the Center panel of Figure 13 does
not change much under a small arbitrary modification of its initial conditions
on past null infinity; but it is drastically modified if we modify its final
conditions on future null infinity. This is intuitively simple to grasp: if we
sit on future null infinity and look back towards the hole, we see a black
disk. This is the final condition. A slightly perturbed final condition
includes the possibility of seeing radiation arriving from this disk. This is
impossible in the spacetime of the Center panel of Figure 13, because of the
huge red shift of the radiation moving next to the horizon, but it is possible
in the Left spacetime, because the radiation may have crossed over from the
other asymptotic region.

The same is true for a white hole, reversing the time direction. In the
spacetime depicted in the Right panel, with some radiation, there is
necessarily a dark spot in the _incoming_ radiation from past null infinity.
If we perturb this configuration, and add some incoming radiation in this dark
spot, the evolution generically gives the spacetime of the Left panel.

Physically, what happens is that this radiation moves along the horizon, is
blue shifted, can meet radiation coming out of the white hole and this is more
mass that $m$ at a radius $r\sim 2m$: it is mass inside its Schwarzschild
radius. At this point the region is trapped, and a black hole forms.
Consequently the evolution of the perturbed initial conditions yields the
spacetime of the Left, not the one of the Right: the white hole is unstable
and decays into a black hole. This is the standard “instability of white
holes” in classical GR.

###  III.5 Planckian Remnants

How does the instability discussed in the last paragraph affect the remnants
formed at the end of a black hole evaporation?

As observed in Bianchi2018e , the wavelength of the perturbation needed to
trigger the instability must be smaller than the size of the hole. To make a
Planck size black hole unstable, therefore, we would need trans-Planckian
radiation, and this is likely not be allowed by quantum gravity.

But let nevertheless assume that a Planck-scale white hole is unstable. Its
decay mode, as explained above, is into a Planck-scale black hole. Using the
notation introduced above, this is the process

$|m_{o},m\rangle_{W}\xrightarrow[\text{instability}]{}|m_{o},m\rangle_{B}.$
(34)

The quantum gravity process discussed earlier in the paper is

$|m_{o},m\rangle_{B}\xrightarrow[\text{tunnelling}]{}|m_{o},m\rangle_{W}.$
(35)

In quantum mechanics, two such processes imply that a system free to loose
energy into radiation will settle on its lowest available energy eigen-state,
which will be a superposition of the two states, of the form

$|m_{o},m\rangle=\alpha|m_{o},m\rangle_{W}+\beta|m_{o},m\rangle_{B}.$ (36)

That is, the remnant will settle into a superposition of black and white
hole.222The possibility of oscillation between black and white hole states has
been also considered in Barcelo:2015uff ; Barcelo2016 ; Garay2017 . In these
works, the tunnelling probability when $m\gg m_{P\ell}$ was estimated to be
large on the basis of a black-hole lifetime computation giving $\tau_{BH}\sim
m_{o}$. In Christodoulou2018 , this computation has been rather reinterpreted
as giving the duration of the tunnelling transition itself.

Since the black and white hole are indistinguishable from the exterior, this
superposition has no effect on the exterior. It is only a quantum spread of
the geometry on and just around the horizon. In the next section we describe
it more precisely.

Minimal energy, for the horizon, means minimal area. However, vanishing area
means flat space, namely total dissipation. The transition to this state is
likely to be strongly suppressed by the arguments in the last section.
Intuitively, this is because it takes time to re-absorb the huge interior, and
its entropic content. Hence, the system will settle for a long time on the
lowest _nonvanishing_ eigenstate of the area.

LQG predicts the area to be quantized Rovelli1993c ; Rovelli1994a . Its
eigenvalues are

$A_{min}=\frac{8\pi\gamma\hbar G}{c^{3}}\sqrt{j(j+1)}.$ (37)

The lowest non vanishing eigenvalue, or “area gap”, is given by
$j=\frac{1}{2}$. Given the relation between area and mass of the horizon, this
gives the mass (1).

That is, LQG predicts that black holes end up as long living particles with a
mass of a few micro-grams.

Once a remnant has attended this minimal size, it is semi-stable. It can still
radiate away its energy, and dissipate into flat space. But this last
transition

$|m_{o},m_{P}\rangle\rightarrow|0\rangle$ (38)

is strongly suppressed and therefore takes a long time, at least of order
$m_{o}^{4}$.

The existence of there objects contradicts the intuition of many particle-
physics or string string theorists. This is because in flat space physics, a
small volume with a small energy can contain only a finite number of different
states. This is what prevents the blackbody UV catastrophe. Intuitively, to
have many states we need many particles, and to have many particles with small
total energy we need to have them with long wavelength, but this is not
available in the volume contained within a small sphere, because this has a
small volume. This is the basis of the intuition of many particle-physics and
string theorists.

In general relativistic physics, however, these constraints do not hold,
because on a curved geometry an arbitrarily large volume can be enclosed into
an arbitrary small sphere. Hence an arbitrarily small horizon can enclose any
number of different states, limited only by the maximal size $m_{o}$ of the
parent black hole and by the finiteness of the time during which the black
hole interior have had the opportunity to grow.

###  III.6 Remnants and quantum spread of the extrinsic curvature

Let us look more closely at the physics of the remnants. Consider the geometry
of a black hole in coordinates where (unlike Schwarzschild) $t=constant$
surfaces properly cut the horizon at different moments. In Lemaitre time, for
instance,

$ds^{2}=-dt^{2}+(dr+\sqrt{2m/r}dt)^{2}+r^{2}d\Omega^{2}.$ (39)

The horizon $r=2m$ has area $A=16\pi m^{2}$. The variable conjugate to the
area is the extrinsic curvature, which has a single (radial-radial) component
$k=a/m^{2}$, where $a$ here is a numerical constant. In quantum theory, $A$
and $k$ are conjugate variables and cannot both be sharp. We expect a
Heisenberg relation of the form

$\Delta A\Delta k>\hbar G.$ (40)

Classical black holes are described by semiclassical quantum states that are
peaked on large values of $A$ and small values of $k$, with $\Delta A$ and
$\Delta k$ satisfying the Heisenberg relations but all relative spreads being
small. Now consider an isolated black hole that radiates away all its energy,
but is stuck on the minimal non vanishing area because of the suppression of
the transition to Minkowski due to its large interior. Then the system settles
on an _eigenstate_ of $A$. This is the common situation of all quantum system,
left alone and free to equilibrate by radiating away energy: they settle on
their minimal reachable energy eigenstate. An eigenstate of the area will have
$k$ maximally spread. Therefore we expect Planck scale remnants to be in
states where $k$ is maximally spread. Now consider a white hole. Its metric is
easilly obtained time reversing the black hole metric, that is

$ds^{2}=-dt^{2}+(dr-\sqrt{2m/r}dt)^{2}+r^{2}d\Omega^{2}.$ (41)

The area of the horizon is the same as the black hole, but $k$ has opposite
sign. An eigenstate of $A$, having $k$ maximally spread does not distinguish
between a black hole and a white hole. Hence remnants must be superpositions
between the two.

###  III.7 There is no information paradox

We close this part of the paper by discussing the _vexata questio_ of the
black hole information paradox, at the light of the above results. This
Section follows closely Rovelli2017a  and Rovelli2019a .

The black hole information problem, as formulated by Don Page in 1993
Page1993a , regards the physics of the spacetime region _before_ the quantum
gravity onset. Figure 15 is the Carter-Penrose diagram of a spherically
symmetric spacetime geometry around a collapsing star, on which there is an
evolving quantum field $\phi$. The geometry takes into account the back-
reaction of the Hawking radiation of the field. The collapsed matter
distribution generates a trapped region: the black hole. The boundary of this
region is the (trapping) horizon. In the limit in which we disregard the back-
reaction of the Hawking radiation, the horizon is null, but taking back-
reaction into account, it is time-like. The notion of _event_ horizon is not
defined, because no assumption is made about the distant future, which depends
on quantum gravity and is not relevant. Considerations on this region alone
are sufficient for Page’s argument for the information problem.

Here is Page’s key observation. Consider a sphere $S$ on the horizon at
retarded time $u$ (see Figure 15). The region of future null infinity
preceding the time $u$ receives the Hawking radiation emitted until the
horizon has reached $S$. Consider the case where the black hole is “old” at
$S$, namely the area $A$ of $S$ is much smaller than the the initial horizon
area $A_{0}$ at $S_{0}$. The Hawking radiation arriving at future infinity
around $u$ is in a mixed state. If the initial state of the field was pure and
evolution is unitary, this radiation must be correlated with something else:
with what?

There are two reasonable possibilities:

(a) it is correlated with degrees of freedom inside the horizon;

(b) it is correlated with degrees of freedom outside the horizon. In
particular, late Hawking quanta may be correlated with early Hawking quanta.

A part of the theoretical community has got convinced that the correct answer
must be (b) because (a) is ruled out by the following argument by Don Page,
based on a specific assumption. Let $N$ be the number of states of a black
hole with which external degrees of freedom can be entangled at some given
time and let

$N_{BH}=e^{A/4}.$ (42)

where $A$ is the area of the horizon at that time. Consider the following

${\rm Assumption:}\ \ \ N\sim N_{BH}.$ (43)

Then, necessarily (a) is ruled out, because for sufficiently late times $A$ is
small and therefore $N$ is insufficient for purifying the Hawking radiation.

But is the assumption justified?

The thermodynamic interaction between a black hole and its surroundings is
well described by treating the black hole as a system with $N_{BH}=e^{A/4}$
(orthogonal) states, where $A$ is the horizon area. But a system can have many
more degrees of freedom than those determining its thermodynamic interactions.
For instance, this is the case when some degrees of freedom are physically
constrained from acting back on the exterior, as is precisely the case for a
black hole. It is only for ergodic systems that the von Neumann entangled
entropy (which purifies the external state) must be equal or smaller than the
thermodynamic entropy, and certainly a black hole is not ergodic. So, there is
definitely no compelling reason based on known physics for assuming (43).

Some theoretical hypotheses, like AdS-CFT or some String formulations elevate
(43) into a postulate. Juan Maldacena has called this assumption a “central
dogma”, and, like all dogmas, it can well be false. As many dogmas, this one
as well leads to bizarre consequences, like firewalls on the horizon
Almheiri2013a ; Marolf2017a .

The possibility that $N\gg N_{BH}$ is supported by the fact that according to
classical GR the interaction between a black hole and its surroundings is
entirely determined by what happen in the vicinity of the horizon, not by the
interior. The number $N_{BH}$ is likely to count only horizon states that
(before the transition) can be distinguishable from the exterior. These are
“surface” states. On the other hand, $N$ counts also states that can be
distinguished by local observables _inside_ the horizon. These do not
contribute to the thermodynamical entropy $S={A/4}$, but can contribute to the
von Neumann entropy, which can therefore remain high even when $A$ shrinks.

These generic considerations can be sharpened. The fact that a blackhole can
have more states than $N_{BH}$, in facts, follows from elementary
considerations of causality.

Figure 15: The portion of spacetime relevant for the information ‘paradox’.
Figure 16: The (lowest part) of the conformal diagram of a gravitational
collapse. The clear grey region is the object, the dotted line is the horizon,
the thick upper line is the singularity, the dark upper region is where
quantum gravity effect may become relevant (this region play no role in this
paper.) The two Cauchy surfaces used in the paper are the dashed lines.

To show this, consider a gravitationally collapsed object and let $\Sigma_{1}$
be a Cauchy surface that crosses the horizon but does not hit the singularity,
see Figure 16. Let $\Sigma_{2}$ be a later similar Cauchy surface and $i=1,2$.
Let $A_{i}$ be the area of the intersection of $\Sigma_{i}$ with the horizon.
Assume that no positive energy falls into the horizon during the interval
between the two surfaces. Let quantum fields live on this geometry, back-
reacting on it WaldCurved . Finally, let $\Sigma_{i}^{in}$ be the (open)
portions of $\Sigma_{i}$ inside the horizon.

Care is required in specifying what is meant here by ‘horizon’, since there
are several such notions (event horizon, trapping horizon, apparent horizon,
dynamical horizon…) which in this context may give tiny differences in
location. For precision, by ‘horizon’ we mean here the event horizon, if this
exist. If it doesn’t, we mean the boundary of the past of a late-time
spacelike region lying outside the black hole (say outside the trapping
region). With this definition, the horizon is light-like.

Because of the back-reaction of the Hawking radiation, the area of the horizon
shrinks and therefore

$A_{2}<A_{1}.$ (44)

Now consider the evolution of the quantum fields from $\Sigma_{1}$ to
$\Sigma_{2}$. We are in a region far away from the singularity and therefore
(assuming the black hole is large) from high curvature. Therefore we expects
conventional quantum field theory to hold here, without strange quantum
gravity effects, at least up to high energy scales. Since the horizon is
light-like, $\Sigma_{1}^{in}$ is in the causal past of $\Sigma_{2}^{in}$. This
implies that any local observable on $\Sigma_{1}^{in}$ is fully determined by
observables on $\Sigma_{2}^{in}$. That is, if ${\cal A}_{i}$ is the local
algebra of observables on $\Sigma_{i}^{in}$ then ${\cal A}_{1}$ is a
subalgebra of ${\cal A}_{2}$:

${\cal A}_{1}\subset{\cal A}_{2}.$ (45)

Therefore any state on ${\cal A}_{2}$ is also a state on ${\cal A}_{1}$ and if
two such states can be distinguished by observables in ${\cal A}_{1}$ they
certainly can be distinguished by observables in ${\cal A}_{2}$ as the first
are included in the latest. Therefore the states that can be distinguished by
${\cal A}_{1}$ —which is to say: on $\Sigma_{1}^{in}$— can also be
distinguished by ${\cal A}_{2}$ —which is to say: on $\Sigma_{2}^{in}$.
Therefore the distinguishable states on $\Sigma_{1}^{in}$ are a subset of
those in $\Sigma_{2}^{in}$. How many are them? Either there is an infinite
number of them, or a finite number due to some high-energy (say Planckian)
cut-off. If there is an infinite number of them, then immediately the number
of states distinguishable from inside the black hole is larger that $N_{NB}$,
which is finite. If there is a finite number of them, then the number $N_{2}$
of distinguishable states on $\Sigma_{2}^{in}$ must be equal or larger than
the number $N_{2}$ of states distinguishable on $\Sigma_{1}^{in}$, because the
second is a subset of the first. That is

$N_{2}\geq N_{1}.$ (46)

Comparing equations (44) and (46) shows immediately that it is impossible that
$N_{i}=e^{A_{i}/4}$, as the exponential is a monotonic function.

This follows from only assuming the validity of quantum field theory in
regions of low curvature. In other words, the “dogma” (43) necessarily
violates known physics in a region where we have no reasons to assume that it
should fail. In our opinion, this is not good scientific method.

The conclusion is that the number of states distinguishable from the interior
of the black hole must be different from the number $N_{BH}=e^{A/4}$ of the
states contributing to the Bekenstein-Hawking entropy. Since the second is
shrinking to zero with the evaporation, the first must overcome the second at
some point. Therefore in the interior of a black hole there are more possible
states than $e^{A/4}$.

The physical interpretation of the conclusion is simple: the thermal behaviour
of the black hole described by the Bekenstein-Hawking entropy $S=A/4$ is
determined by the physics of the vicinity of the horizon.

A vivid manifestation of the fact that in classical GR the effect of a black
hole on its surroundings is independent from the black hole interior is in the
numerical simulations of black hole merging and radiation emission by
oscillating black holes: in writing the numerical code, it is routine to cut
away a region inside the (trapping) horizon: it is irrelevant for whatever
happens outside! This is true in classical GR, and there is no compelling
reason to suppose it to fail if quantum fields are around.

The idea of interpreting of $S_{BH}$ as determined by the number of states of
near-surface degrees of freedom, and not interior ones is of course not a new
idea. It has a long history York ; Zurek ; Wheeler1990 ; THooft1990 ;
Susskind1985 ; Frolov1993 ; Larsen1996 . See in particular Rovelli1996d ,
Strominger:1997eq  in support of this idea from two different research camps,
loops and strings.

Importantly, this conclusion is _not_ in contrast with the various arguments
leading to identify Bekenstein-Hawking entropy with a counting of states. To
the opposite, evidence from it comes from the membrane paradigm Thorne1986
and from Loop Quantum Gravity Rovelli1996a ; Rovelli1996d ; Ashtekar:1997yu ;
Perez2017a , which both show explicitly that the relevant states are surface
states, but also from the string theory counting Strominger:1996sh ;
HorowitzStrominger , because the counting is in a context where the relevant
state space is identified with the scattering state space, which could be
blind to interior observables. For a classic discussion on different
viewpoints about these alternatives, see Jacobson2005

In conclusion, if there are more states available in a black hole than
$e^{A/4}$, then Page argument for the information loss paradox fails. States
purifying the Hawking radiation are inside the hole even when the horizon
shrinks. The information can later escape from the remnant emission (see
below).

Notice that in popular accounts (for a recent one, see for instance Wuthrich
), three assumptions are erroneously said to be proven incompatible: (i)
unitarity of the quantum evolution, (ii) equivalence principle (absence of
firewalls), and (iii) quantum field theory on curved spacetimes. This is
wrong. It is only the further assumption of the dogma, that leads to problems.

##  IV Elements of Phenomenology

Measuring quantum gravity effects is notoriously difficult Liberati2011 .
Still, some intriguing possibilities of observation are opened by the scenario
described above. The first is the possibility that the remnants described
above are the constituents of, or contribute to, dark matter Rovelli2018f . In
particular, remnants forming dark matter could be produced by primordial (or
pre-big bang) black holes. Direct detection of these remnants has been studied
in Perez2023 . A number of possible astrophysical implications of this
scenario or variants have been tentatively explored in Vidotto2016 ;
Rovelli2017f ; Barrau2017 ; Raccanelli2017 ; Vidotto2018a ; Rovelli2018f ;
Vidotto2018 ; Barausse2020 ; barrau2021closer .

###  IV.1 Dark Matter

Remnants are a dark matter candidate that does not require exotic assumptions
of new forces, or particles or corrections to the Einstein equations, or
physics beyond the standard model. It only requires general relativity and
quantum theory to hold together.

The possibility that remnants of evaporated black holes of primordial origin
could form a component of dark matter was suggested by MacGibbon
J.H.MacGibbon1987  thirty years ago and has been explored by many authors
Barrow1992 ; Carr1994 ; Liddle1997a ; Konoplich1999 ; Khlopov1999 ;
barrau2003improved ; Chen2003 ; Chen2004 ; Khlopov2007a ; Nozari2008 ;
Dymnikova2015 . Since there are no strong observational constraints on this
potential contribution to dark matter Carney2018a , the weak point of the
scenario has been, until now, the question of the physical nature of the
remnants. The scenario discussed in these notes has changed the picture:
conventional physics provides a candidate for remnants.

###  IV.2 Direct detection

Figure 17: A particle of mass $m$ in a superposition state with separation
$\epsilon$. The DM particle passes by with velocity $v$ and a closest approach
distance $d$.

.

A local dark matter density of the order of $0.01M_{\odot}/pc^{3}$ corresponds
to approximately one Planck-scale white hole per each $10.000Km^{3}$. These
objects are presumably moving fast with respect to our local frame, since we
are rotating with the galaxy at hundreds of Km per second, while dark matter
probably isn’t. This gives a very rough estimates of a few Planck scale
particles per $m^{2}$ per year flying by us. Can we detect them?

If the sole interaction of remnants is gravitational, they are very good dark
matter candidates, but for this same reason direct detection using classical
sensing is challenging Carney2018a , due to the extreme weakness of the
gravitational interaction. However, there may be a quantum technology that
could open a window to do the detection. In fact, recent developments in the
area of table-top experiments involving gravity and quantum phenomena (see for
instance Christodoulou2022  for up do date references) open the theoretical
possibility of direct detection of purely-gravitationally-interacting dark
matter particles. A idealized detector where the center of detector mass is
set in a superposition of locations and a more concrete tentative protocol,
which employs Josephson junctions, have been illustrated Perez2023 .

Figure 17 illustrates the conceptual setting. The dark matter particle flies
by the device, passing closer to one of the two locations in which a quantum
particle is quantum split. The different momentum transfer determines a phase
difference between the two branches, that can be picked up by interferometry,
or as a current change in an array of Josephson junctions. See Perez2023  for
details.

###  IV.3 Cosmological implications

Remnants could be produced by primordial black holes formed in the early
universe. The lower bound of the lifetime of the remnants is of the order or
lower to $m_{o}^{4}$. If we assume that lifetime is precisely of the order of
the lower bound $m_{o}^{4}$ (as mentioned, this may not be correct), then for
these objects to be still present now we need their lifetime to be larger or
equal than the Hubble time $T_{H}$, that is

$m_{o}^{4}\geq T_{H}.$ (47)

On the other hand, we expect these to be produced by evaporated black holes,
therefore the lifetime of the black hole must be shorter than the Hubble time.
Therefore

$m_{o}^{3}<T_{H}.$ (48)

This gives an estimate on the possible value of $m_{0}$:

$10^{10}gr\leq m_{o}<10^{15}gr.$ (49)

These are the masses of primordial black holes that could have given origin to
dark matter present today in the form of remnants. Their Schwarzschild radius
is in the range

$10^{-18}cm\leq R_{o}<10^{-13}cm.$ (50)

According to primordial black hole formation theory, black holes of a given
mass could have formed when their Schwarzschild radius was of the order of the
horizon. Remarkably, the horizon was presumably in this range at the end of
inflation, during or just after reheating. A preliminary phenomenological
analysis of this particular scenario was carried out in Barrau2014 .

If the lifetime can be longer, constraints are less stringent. Furthermore, an
intriguing possibility opens up. Remarkably, the strength of the interaction
of such particles, combined with the assumption of a sufficiently hot big
bang, leads to a density of these objects at decoupling whose order of
magnitude is compatible with the present dark matter density Barrau2019 ;
Amadei2022 .

###  IV.4 Erebons

An alternative possibility for the generation of remnants is that they were
formed in a contracting phase before the current expanding one, in a big
bounce scenario (for a review of classical and quantum bouncing cosmologies,
see Brandenberger2016 ; Agullo2016  and references therein).

The possibility that black holes could live across the big bounce and
represent a component of dark matter has been considered in Carr2017 . Roger
Penrose has coined the name _erebons_ , from the Greek god of darkness Erebos,
to refer to matter crossing over from one eon to the successive one
Penrose2017  in his cyclic cosmology Penrose2012 . Large black holes
evaporated before the bounce could have given rise to a population of Planck-
size white holes remnants that has crossed the bounce and formed what we see
today as dark matter. For this to happen, their density should have been
sufficient to balance the huge dilution in an eventual inflationary phase.

An intriguing aspect of this scenario is the speculative idea that it might
addresses the apparent low-entropy of the initial cosmological state
Rovelli2018i . This can be consistent and can represent a concrete realisation
of the perspectival interpretation of entropy suggested in Rovelli:2015 . If
the cosmos at the big bounce was finely dotted by white holes with large
interiors, then the gravitational field was not in the very improbable low
entropy homogeneous or nearly homogeneous configuration. It was in a high
entropy crumpled configuration. But being _outside_ all white holes, we are in
a special place, and from the special perspective of this place we see the
universe under a coarse grain which defines an entropy that was low in the
past.

###  IV.5 Modeling remnants emission

Finally, remnants must themselves emit, although very softly, in order to
release the information that was trapped with the in-falling Hawking radiation
and dissipate (as white holes do).

Classically, this a very low power emission. Quantum mechanics, it is given by
a weak decay probability, akin radioactivity. This process was tentatively
modelled in Kazemian2022a , which we summarize here.

Say a black hole of initial mass $m_{o}$ evaporates via Hawking evaporation,
leaving a remnant of Planckian mass which contains an amount of information
sufficient to purify its Hawking radiation, namely of order

$S\sim\frac{A}{4}=4\pi m_{o}^{2}$ (51)

Here $A$ is the area of the horizon at formation. Information is emitted in
form of radiation. Since the radiation is emitted radially, it can be modeled
as a uniform one-dimensional gas of photons. Assume for simplicity that the
radiation emitted emitted by the surface of the remnant during the lifetime
$\tau$, is in thermal equilibrium Marolf2017a . At the end of the remnant
lifetime the radiation covers a length $L=\tau$. The energy $E$ available for
this gas is only that of the mass of the remnant, which is of the order of the
Planck mass, namely unit in natural units.

$E\sim 1,$ (52)

while its total entropy, needed to purify the Hawking radiation is (51).
Entropy $S$ and energy $E$ of a 1d photon gas of temperature $T$ in a space of
length $L$ are skobelev2013entropy

$S=\frac{2\pi}{3}LT,\ \ \ \ E=\frac{1}{6}LT^{2},$ (53)

which gives

$L=6m^{4},\ \ \ \ T=\frac{1}{m^{2}},$ (54)

and a number of quanta (say photons) emitted

$N_{\gamma}\sim{m^{2}}.$ (55)

The life-time of a white hole would be equal to time required for the photons
to travel a distant $L$. We therefore have

$\tau_{W}\sim 6m^{4}$ (56)

as already mentioned, while the temperature of the white hole emission is be
much lower than the initial temperature $\sim 1/m$ of the Hawking temperature
of the parent black hole.

A population of black holes formed at a time $t=0$, with mass $m$ and
uniformly distributed in space that evaporate around time $\tau_{B}\sim m^{3}$
as predicted by Hawking radiation theory, and survive as white hole remnants
for a time $\tau_{W}$ as in (56) emits a steady radiation between times
$\tau_{B}$ and $\tau_{B}+\tau_{W}$. For $\tau_{B}<t<\tau_{B}+\tau_{W}$ an
observer would observe the radiation density would evolve in time as

$\rho(t)\left\\{\begin{array}[]{lll}=0&{\rm for}&t<m^{3},\\\
=\big{(}\frac{t-\tau_{B}}{\tau_{W}-\tau_{B}}\big{)}\Omega&{\rm
for}&m^{3}<t<6m^{4},\\\ =\Omega&{\rm for}&t>6m^{4}.\end{array}\right.$ (57)

In other words, the emission process is a steady (linear in time)
transformation of dust into radiation, on a $m^{4}$ timescale. See Figure 18.

Figure 18: Background white hole radiation as a function of time. The solid
black line represents a classical linear emission while the dashed red line
represents a quantum emission.

Quantum mechanics alters this picture a bit. A continuous energy emission as
the one described above would imply a continuously decreases of the white hole
horizon area, below the Planck area. But this is not permitted by LQG, because
area is quantized.

Rather, a remnant with near-Planckian mass and area must make a single quantum
leap into radiation, as in conventional nuclear radioactivity, where emission
is realised by individual quantized quanta governed by a probability
distribution Martin2019 .

In other words, in first order perturbation theory the only allowed transition
is the emission of the entire Planck energy of the remnant.

In the language of quantum field theory, this is given by a vertex between the
remnant and a large number of low energy quanta. A vertex coupling a remnant
to a single or a few photons, indeed, is forbidden by conservation of
information (unitarity), because a few photons do not have enough degrees of
freedom to match the large number of quantum numbers describing the white hole
interior. Few photons cannot carry the entire information that can be stored
in the remnant. Hence the transition must be of the form
$remnant\to\gamma_{1}...\gamma_{n}$ to a large number of low energy photons:

The number of photons emitted by a single remnant is given in (55). This
conclusion is relevant in view of an old objection to the remnant scenario,
that contributed to its abandonment in the Nineties. The objection was that
the large number of remnant internal states would make them too easy to
produce in particle physics experiments. Here we see why that conclusion was
to quick. The effective vertex responsible for a remnant production would be

in order to create a long living Planck size remnant. If the number of photons
was small, these could be high energy, but the remnant produced would
correspond to a remnant whose parent is a black hole of Planckian size, but
short lived. The process would not be distinguishable by the standard collapse
predicted by conventional quantum gravity. To produce an actual long living
remnant, on the other hand, we need $m$ to be large, and hence we would need
to focus a large number of low energy photons.

In the cosmological standard model, primordial black holes may have formed at
reheating. To get a sense of the characteristic of the diffuse radiation
remnants may emit, one can estimate its parameter in this simplest case
Kazemian2022a . This model is entirely determined by a single parameter or
order of unity, that can be taken to be

$x=\log_{10}(m/m_{Pl})\in[15,20].$ (58)

The results for mass, frequency and density of the emitted radiation are

$\displaystyle m$ $\displaystyle=$ $\displaystyle 10^{x-5}gr,$ (59)
$\displaystyle\nu$ $\displaystyle=$ $\displaystyle 10^{-2x+32}Hz$ (60)
$\displaystyle\rho_{rad}$ $\displaystyle=$
$\displaystyle\sinh\bigg{(}\frac{10^{61}-10^{3x}}{10^{4x}-10^{3x}}\bigg{)}\rho_{rem}$
(61)

This is a mass range

$10^{10}gr<m<10^{15}gr,$ (62)

and a frequency range

$10^{14}Hz>\nu>10^{4}Hz.$ (63)

Thus, a diffused radiation at frequency $\nu$ and density $\rho_{rad}$ can
witness a cold dark component formed by white hole remnants, descending from
primordial black holes of mass

$m=10^{x}m_{Pl}$ (64)

and with density

$\rho_{rem}=\sinh^{-1}\\!\\!\left(\frac{1-10^{61}\cdot 10^{-3x}}{1-6\cdot
10^{-x}}\right)\rho_{rad}$ (65)

where $x$ can be measured directly from the frequency of the diffused
radiation:

$x=-\frac{1}{2}\ \log_{10}\frac{\nu}{\nu_{P}l}$ (66)

Other cosmological scenarios, such as big bounce or matter bounce scenarios,
remnants might also –and perhaps better– accommodate remnants as a component
of dark matter.

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