# Effective models of non-singular quantum black holes

M. Cadoniab, M. Oiab, A. P. Sannaab
a _Dipartimento di Fisica, Università di Cagliari_
Cittadella Universitaria, 09042 Monserrato, Italy

b _I.N.F.N, Sezione di Cagliari_
Cittadella Universitaria, 09042 Monserrato, Italy

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate how the resolution of the singularity problem for the
Schwarzschild black hole could be related to the presence of quantum gravity
effects at horizon scales. Motivated by the analogy with the cosmological
Schwarzschild-de Sitter solution, we construct a broad class of non-singular,
static, asymptotically-flat black-hole solutions with a de Sitter (dS) core,
sourced by an anisotropic fluid, which effectively encodes the quantum
corrections. The latter are parametrized by a single length-scale $\ell$,
which has a dual interpretation as an effective "quantum hair" and as the
length-scale resolving the classical singularity. Depending on the value of
$\ell$, these solutions can have two horizons, be extremal (when the two
horizons merge) or be horizonless exotic stars. We also investigate the
thermodynamic behavior of our black-hole solutions and propose a
generalization of the area law in order to account for their entropy. We find
a second-order phase transition near extremality, when $\ell$ is of order of
the classical Schwarzschild radius $R_{\text{S}}$. Black holes with $\ell\sim
R_{\text{S}}$ are thermodynamically preferred with respect to those with
$\ell\ll R_{\text{S}}$, supporting the relevance of quantum corrections at
horizon scales. We also find that the extremal configuration is a zero-
temperature, zero-entropy state with its near-horizon geometry factorizing as
AdS${}_{2}\times$ S2, signalizing the possible relevance of these models for
the information paradox. Finally, we show that the presence of quantum
corrections with $\ell\sim R_{\text{S}}$ have observable phenomenological
signatures in the photon orbits and in the quasi-normal modes (QNMs) spectrum.
In particular, in the near-extremal regime, the imaginary part of the QNMs
spectrum scales with the temperature as $c_{1}/\ell+c_{2}\ell
T_{\text{H}}^{2}$, while it goes to zero linearly in the near-horizon limit.
Our general findings are confirmed by revisiting two already-known models,
which are particular cases of our general class of models, namely the Hayward
and gaussian-core black holes.

###### Contents

  1. 1 Introduction
  2. 2 Unified description of space-time and matter inside a black hole
  3. 3 Spherically symmetric solutions sourced by anisotropic fluids
    1. 3.1 Equation of state and energy conditions
    2. 3.2 Absence of singularity and behavior near $r=0$
    3. 3.3 Asymptotic flatness and behavior at $r\to\infty$
  4. 4 A general class of non-singular quantum black-hole models
    1. 4.1 Energy conditions
    2. 4.2 Extremal limit
    3. 4.3 Black hole thermodynamics
    4. 4.4 Phase transition
    5. 4.5 Null geodesics and photon orbits
    6. 4.6 Quasi-Normal Modes spectrum in the eikonal approximation
  5. 5  A minimal model: the Hayward black hole
    1. 5.1 Thermodynamics and phase transition
    2. 5.2 Null geodesics and quasi-normal modes in the eikonal limit
  6. 6 Quantum black holes with gaussian cores
    1. 6.1 Quantum black hole regimes
    2. 6.2 Thermodynamics and phase transition
    3. 6.3 Null geodesics and photon orbits
    4. 6.4 Quasi-normal modes spectrum in the eikonal limit
  7. 7 Conclusions
  8. A Geodesic completness of the spacetime

##  1 Introduction

Since the discovery of the Schwarzschild solution, the presence of a
singularity inside black holes, together with the initial cosmological one,
has represented a serious challenge to our current understanding of the
fundamental laws of physics. This problem became even more serious after the
groundbreaking Penrose and Hawking singularity theorems [1, 2]. They proved
incontrovertibly that, under a set of a few, very general and physically
motivated assumptions (the validity of the weak energy condition and either
global hyperbolicity or the validity of the strong energy condition), these
space-time singularities are unavoidable, at least in the classical general
relativity (GR) framework. Despite this, it is conjectured that these
singularities are always hidden behind a causal barrier, the event horizon,
which prevents outside observers from seeing them and the theory from
completely loosing its predictive power [3]. Semiclassical effects, like black
hole evaporation [4], seem however to bring the singularity problem back on
the table, as the final steps of the evaporation process, where the
singularity role should be most prominent, are still poorly understood.

Although it is in principle solvable already in the classical GR framework by
relaxing some assumptions of Penrose’s theorem and constructing non-singular
effective models (see, e.g. Refs. [5, 6, 7, 8, 9, 10, 11]; for models with
non-linearly coupled electromagnetic fields, see [12, 13, 14, 15, 16, 17]) the
singularity problem calls for the need of a quantum description of
gravitational interactions. The most widely adopted approach in the past has
been to assume that these quantum corrections should only be relevant at
Planck scale, $\ell_{\text{P}}=\sqrt{G}\sim 10^{-35}\ \text{m}$ 111We adopt
natural units, $c=\hbar=1$. We will use $G$ or $\ell_{\text{P}}^{2}$
interchangeably. [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32].
Quantum gravity effects should become important only when the Compton length
of a pointlike mass $M$ becomes comparable with its Schwarzschild radius,
$R_{\text{S}}=2GM$. Thus, they should be irrelevant as long as gravitational
interactions at Planck-scale distances are not considered, like the final
stages of the evaporation of black holes, the behavior of space-time near
their central singularity or the initial phases of the evolution of the
universe.

In recent times, however, there have been many indications supporting the
possibility of having relevant quantum gravity effects even at scales much
larger than $\ell_{\text{P}}$, i.e. at horizon or cosmological scales. At the
black-hole level, this new perspective gains motivation from different
approaches: the firewall paradox [33], which triggered several recent advances
in tackling the black-hole information puzzle (islands and replica wormholes
[34, 35, 36, 37], non-local modifications of effective field theory [38, 39,
40], fuzzball proposal [41, 42, 43]); the emergent gravity and corpuscular
gravity scenarios [44, 45, 46, 47, 48, 49, 50], in which a black hole is
considered as a coherent state of a large number of gravitons of typical
wavelength $\sim R_{\text{S}}$ [51, 52, 53]; finally, the quasi-normal modes
(QNMs) spectrum of the Schwarzschild black hole, whose description is
consistent with that of an ensemble of oscillators with typical frequency
$\omega\sim 1/R_{\text{S}}$ [54, 55]. Further evidence came from the galactic
and cosmological framework, where deviations from Newtonian dynamics and the
evolution of dark energy can be interpreted in terms of long-range quantum
gravity effects, described by an exotic source of Einstein’s equation in the
form of an anisotropic fluid [44, 56, 57, 58, 59, 60, 61].

The possibility of having quantum-gravity effects operating at black-hole
horizon scales is also extremely interesting from a phenomenological point of
view. These effects are expected to be encoded in the QNMs spectrum and to be
detected by the next generation of gravitational wave (GW) detectors, like the
Einstein Telescope (ET), in the ringdown phase of two compact objects merging
to form a single black hole. In some particular cases, a manifest signature
could be the presence of echoes in the GW signal [62, 63, 64].

The starting idea of this paper is that the resolution of the singularity
problem could be related to the presence of quantum gravity effects at horizon
scales. This is somehow natural because we expect quantum effects to be at
work both in the smearing of the classical black-hole singularity and in
generating an effective quantum hair at horizon scales. We parametrize the
smearing of the classical singularity with a length-scale $\hat{L}$, whereas
the quantum hair is represented by an extra length-scale $\ell$. We assume
that this smearing is sourced by an exotic form of matter having the form of
an anisotropic fluid, which should give an effective description of quantum
gravity effects. The analogy with galactic dynamics, where an infrared (IR)
scale $R_{0}=\sqrt{R_{\text{S}}L}$ is generated out of $R_{\text{S}}$ and the
size of the cosmological horizon $L$ [44], now suggests that, similarly,
$\ell$ can be interpreted as an IR scale generated from $R_{\text{S}}$ and
$\hat{L}$, for instance by the simple relation

$\ell=R_{\text{S}}^{a}\hat{L}^{b}$ (1)

with $a+b=1$. Thus, the origin of the quantum hair $\ell$ should find
explanation in the multi-scale behavior of gravitational interactions.

Following the cosmological analogy, we can think of a non-singular black hole
as a "reversed" Schwarzschild-de Sitter (SdS) space-time, in which the
external cosmological horizon and the inner Schwarzschild one are
interchanged, and for which the length-scale $\hat{L}$ becomes the de Sitter
(dS) length. In this way, we are motivated to construct a general class of
non-singular, static, asymptotically-flat black-hole solutions with a dS core,
sourced by an anisotropic fluid, which endows the classical Schwarzschild
solution with a quantum hair $\ell$. Extending this similarity with the SdS
case and with the dynamically generated scale $R_{0}$, we will explicitly
prove that $\ell$ is dynamically generated by $R_{\text{S}}$ and $\hat{L}$ by
$\ell\sim R_{\text{S}}^{1/3}\hat{L}^{2/3}$, a relation which should hold in
general for regular models with dS cores.

We find that imposing a regular dS core $a)$ always violates the strong energy
condition in the interior of these objects, and therefore allows us to
circumvent the singularity theorem, and $b)$ depending on the value of the
parameter $\ell$, our non-singular models can have two, one (extremal
configuration) or no horizons. We then proceed by investigating the
implications of the presence an extra parameter $\ell$, assumed to be of the
same order of magnitude as $R_{\text{S}}$, on the thermodynamic properties of
the black hole and on the phenomenology of the models, i.e. on photon orbits
and on the QNMs spectrum.

By using the first law of thermodynamics, we show that the presence of $\ell$
causes deviations from the standard area law. We propose therefore an entropy
formula to generalize the latter. Using this general entropy formula, we also
find that the extremal configuration is a zero-temperature, zero-entropy
state, a behavior drastically different from extremal Reissner-Nordström (RN)
and Kerr black holes. This, together with the fact that the extremal, near-
horizon, geometry factorizes as the tensor product of two-dimensional Anti de
Sitter (AdS2) with a two-sphere, i.e. AdS${}_{2}\times$ S2, indicates that
these regular models could actually be relevant for tackling the information
paradox [65, 66, 67, 68, 35, 69]. By investigating the behavior of the
specific heat and the free energy of the hole, we find a second-order phase
transition near extremality, i.e for $\ell\sim R_{\text{S}}$. In particular,
black holes with $\ell\sim R_{\text{S}}$ are energetically preferred with
respect to those with $R_{\text{S}}\gg\ell$, lending further support to the
possible relevance of quantum corrections at horizon scale.

On the phenomenological side, we find that, for black holes with $\ell\ll
R_{\text{S}}$, deviations from standard results concerning photon orbits and
the QNMs spectrum are negligibly small and not detectable, at least in the
near future. Conversely, black holes with $\ell\sim R_{\text{S}}$ are
characterized by macroscopic deviations from the Schwarzschild behavior, whose
signatures are potentially detectable by the next generation of GW detectors.
In particular, by analytically computing the QNMs spectrum in the eikonal
approximation, we find that, in the near-extremal limit, the imaginary part of
the quasi-normal frequencies scale with the black-hole temperature as
$c_{1}/\ell+c_{2}\ell T_{\text{H}}^{2}$ (with $c_{1,2}$ constants), while in
the near-extremal and near-horizon regimes, it goes to zero, in agreement with
several results in the literature [70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81]. This appears to be a general feature of non-singular black holes, common
also to charged and/or rotating extension of regular models [82, 83, 84].

In the final part of the paper, we check our results by revisiting two
already-known models, namely the Hayward and gaussian-core black-hole metrics,
which represent particular cases of our general class of regular black holes.

The outline of the paper is the following. In section 2 we build up the
grounds for our multi-scale description of gravity by drawing an analogy
between the SdS solution and galactic dynamics from one side, and regular
black-hole models for the other side.

In section 3, we find the exact, most-general, spherically-symmetric static
solution of Einstein’s field equations, sourced by an anisotropic fluid, and
we outline the basic requirements needed to avoid the central singularity. We
then focus on a subclass of such models by choosing a particular equation of
state and analyze the null and strong energy conditions.

In section 4, we select the general class of regular black-hole solutions by
imposing a set of minimal constraints, namely dS behavior in the interior,
asymptotically flatness at infinity and the presence of horizons. We also
study the general thermodynamic behavior of these models, discussing the first
law of thermodynamics and the appearance of the second-order phase transition.
Finally, we investigate photon orbits and the QNMs spectrum in the eikonal
approximation.

In section 5 and section 6 the general discussion is applied and the results
are confirmed by revisiting two previously-proposed regular black-hole models,
the Hayward and the gaussian-core ones, which appear as particular cases of
our general class of models. We finally state our conclusions in section 7.

##  2 Unified description of space-time and matter inside a black hole

As anticipated in the introduction, in the present paper we adopt a
description of gravitational interactions in terms of an effective multi-scale
field theory, characterized by the generations of hierarchically different
length-scales.

This description is natural in the cosmological and galactic context, as
gravity and baryonic matter are characterized by: $1)$ the Planck length
$\ell_{\text{P}}$, $2)$ the size of the cosmological horizon $L$, related to
the cosmological constant by $L=\Lambda^{-1/2}$, and $3)$ the gravitational
radius of a clump of baryonic matter with mass $M$,
$R_{\text{S}}\sim\ell_{\text{P}}^{2}M$. As already mentioned, an intermediate
(mesoscopic) IR length scale in the galactic regime

$R_{0}=\sqrt{R_{\text{S}}L}$ (2)

is dynamically generated from $R_{\text{S}}$ and $L$. At this scale, gravity
deviates from its Newtonian behavior as it is evident from rotational curves
of galaxies. Moreover,$R_{0}$ can be seen as a scale at which long-range
quantum gravity effects become relevant [44, 57, 56, 58]. This scenario allows
for an effective description in the GR framework in terms of an anisotropic
fluid, which can be seen as a two-fluid model of dark energy and matter [57,
56, 58, 59, 60, 61]. The resulting space-time is the SdS solution, in which
dark energy dominates at very large scale. In this regime, we have a
description in terms of the pure dS space-time and a related scale isometry
[85]. When instead clustered matter $M$ is present and becomes non-negligible,
the scale invariance of the dS-background is broken, the quantum scale $R_{0}$
is generated and we have an effective description in terms of the SdS space-
time. The latter is characterized by an internal Schwarzschild-like horizon,
determined by the baryonic mass $M$, and by an external dS horizon, which, for
small $M$, is located at $r=L$. The short-scale regime, instead, is described
by the Schwarzschild solution with a related scale $R_{\text{S}}$, at which
the matter contribution dominates over dark energy. The geometry is
asymptotically dS 222Notice that, in order to make contact with a black hole
space-time, we have to use a static parametrization of the dS geometry..

In the emergent gravity scenario of Ref. [44], these two regimes are assumed
to be endowed with a microscopic description in terms of quantum gravity
degrees of freedom (DOFs) entangled on short-scales (at $r\sim R_{\text{S}}$)
and on cosmological scales ($r\sim L$). Following Refs. [44, 56], the short-
range entanglement is responsible for the holographic horizon-area scaling of
the entropy. The long-range regime is, instead, characterized by the slow
thermalization of IR, long-range interacting, quantum-gravity DOFs. This IR
dynamics is responsible for an extensive, i.e. volume-dependent, contribution
to the entropy. As argued in Ref. [44], the competition between the area- and
volume-laws in the entropy generates a mesoscopic scale $R_{0}$ and an
additional gravitational dark force explaining the deviations from the
Newtonian dynamics at galactic scales. This multiscale description of gravity,
with a "fast scale", $R_{\text{S}}$ and and a "slow scale" $L$, is reminiscent
of thermodynamic systems characterized by a glass transition [44] 333 At short
time scales, glassy systems have properties which cannot be distinguished from
those of crystals: their effective descriptions are identical. However, the
former are characterized by a long timescale behavior, which makes them
completely different from crystals..

Following this line of reasoning, one is led by analogy to use a similar
multiscale description of matter and gravity for the black-hole interior, in
particular to solve the singularity problem. We will consider only macroscopic
black holes, i.e. black holes whose horizon radius is hierarchically larger
than $\ell_{\text{P}}$. The short-distance behavior in the black-hole interior
(near the singularity) is now dominated by the short-scale dynamics of the
emergent space-time DOFs. It is natural to assume that, similarly to the
cosmological case at large scales, here the contribution of matter is
negligible at short scales, where we have an effective GR description in terms
of a pure dS space-time. This regime is therefore characterized by an
ultraviolet (UV) dS length $\hat{L}$, a related cosmological constant
$\hat{\Lambda}=\hat{L}^{-2}$ and scale invariance. This description is fully
consistent with the existence of an UV fixed point, predicted by the quantum-
gravity asymptotic safety scenario (see, e.g. Refs. [86, 87, 88]; for a
resolution of the classical singularity in the asymptotic safety scenario, see
Ref. [89]; for recent results on scale invariance in the core of black holes,
see Ref. [90]). Moreover, the dS behavior of the space-time at short scales is
consistent with volume-law contribution to the entropy.

Introducing baryonic matter $M$ breaks the scale and conformal invariance of
the dS space-time in the black-hole interior. Similarly to the galactic and
cosmological regimes [58], in this case a new quantum scale $\ell$ is
generated in terms of $\hat{L}$ and $R_{\text{S}}$. Using an argument similar
to that of Ref. [44], the generation of $\ell$ can be also explained in terms
of the competition between the short-range, volume contribution and the area-
law-Bekenstein-Hawking contribution to the entropy we have at large distances
from the center, at the Schwarzschild radius $R_{\text{S}}$.

We see therefore that a multiscale description of gravitational interactions
can be adopted both to describe black-holes in a cosmological background and
the interior of asymptotically flat black holes. In the two cases, however,
the horizon positions are reversed. In the latter case, the dS horizon is the
internal one, whereas the matter-determined horizon is the external one. For
this reason, even if we expect $\ell=f(\hat{L},R_{\text{S}})$, this relation
needs not to be the same as that relating $R_{0}$, $L$ and $R_{\text{S}}$ in
eq. 2. Another difference from the cosmological SdS case, is that here we have
the possibility of an external description, i.e. a description of an
asymptotic observer at $r\to\infty$. The latter sees "quantum" deviations from
the Schwarzschild geometry, parametrized by $\ell$. In this respect, it should
be emphasised once again that the relation between the cosmological case,
described by eq. 2, and the black-hole case, described instead by eq. 1, is
that of an analogy. In particular, this prevents one from finding any relation
between the cosmological scales $(L,R_{0})$ and our scales $(\hat{L},\ell)$.

From this perspective, we have a new phase in the black-hole interior, in
which the emergent gravity DOFs and matter should allow for an effective two-
fluids description, i.e. an effective description in terms of an anisotropic
fluid [91]. In the next sections, we will construct a general class of GR
models describing gravity sourced by an anisotropic fluid, which allows for
non-singular black-hole solutions with two event horizons and an internal dS
core.

##  3 Spherically symmetric solutions sourced by anisotropic fluids

Our starting point is GR sourced by an anisotropic fluid. The stress-energy
tensor $T_{\mu\nu}$ appearing in Einstein’s equations $G_{\mu\nu}=8\pi
GT_{\mu\nu}$ will be that pertaining to an anisotropic fluid. Anisotropic
fluids have a long history and have been fruitfully used in several different
contexts in gravitational studies, including compact objects, singular and
non-singular black hole models, cosmology (for an incomplete list, see, e.g.
Refs. [57, 59, 60, 61, 92, 91, 93, 94, 22, 95, 96, 97, 98, 99, 100, 101, 102,
8, 103, 104, 11]).

We consider static, spherically-symmetric solutions of the theory, whose
metric part can be written in the form

$ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}d\Omega^{2};\hskip
14.22636ptd\Omega^{2}=d\theta^{2}+\sin^{2}\theta\ d\phi^{2}.$ (3)

where $\nu(r)$ and $\lambda(r)$ are metric functions, depending on the radial
coordinate $r$ only.

The stress-energy tensor describing the anisotropic fluid can be written as
[92]

$T_{\mu\nu}=\left(\epsilon+p_{\perp}\right)u_{\mu}u_{\nu}+p_{\perp}\
g_{\mu\nu}-\left(p_{\perp}-p_{\parallel}\right)w_{\mu}w_{\nu},$ (4)

where $\epsilon(r)$, $p_{\parallel}(r)$ and $p_{\perp}(r)$ are the energy
density and the radial and tangential pressure components, respectively, while
$u_{\mu}$ and $w_{\mu}$ are 4-vectors satisfying the normalization conditions
$g^{\mu\nu}u_{\mu}u_{\nu}=-1$, $g^{\mu\nu}w_{\mu}w_{\nu}=1$ and
$u^{\mu}w_{\mu}=0$.

The independent Einstein’s field and stress-energy tensor conservation
equations read (the prime denotes derivation with respect to $r$)

$\displaystyle\frac{1-e^{-\lambda}+re^{-\lambda}\lambda^{\prime}}{r^{2}}=8\pi
G\epsilon;$ (5a)
$\displaystyle\frac{e^{-\lambda}-1+re^{-\lambda}\nu^{\prime}}{r^{2}}=8\pi
Gp_{\parallel};$ (5b) $\displaystyle
p^{\prime}_{\parallel}+\frac{\nu^{\prime}}{2}\left(\epsilon+p_{\parallel}\right)+\frac{2}{r}\left(p_{\parallel}-p_{\perp}\right)=0.$
(5c)

Integration of the first equation yields

$e^{-\lambda(r)}=1-\frac{8\pi G}{r}\int\epsilon\ r^{2}dr\equiv
1-\frac{2Gm(r)}{r}$ (6)

where $m(r)$ is the Misner-Sharp (MS) mass

$m(r)\equiv 4\pi\int_{0}^{r}d\tilde{r}\,\tilde{r}^{2}\,\epsilon(\tilde{r}).$
(7)

Using eq. 7, eq. 5b can be recast in the more useful form

$\frac{\nu^{\prime}}{2}=\frac{4\pi
Gp_{\parallel}r^{3}+Gm}{r\left(r-2Gm\right)}.$ (8)

The system (5a)-(5c) is not closed. In order to determine the solution
unambigously, we must support eqs. 5a and 5c with two further equations. The
simplest and physically natural way to close the dynamical system is to
provide: $1)$ a barotropic equation of state (EoS) for the radial pressure
$p_{\parallel}=p_{\parallel}(\epsilon)$ and $2)$ the matter density profile
$\epsilon(r)$. In the following, we will fix the equation of state and the
matter density profile by imposing absence of singularities, Schwarzschild
behavior at $r\to\infty$ and using the analogy with cosmology discussed in
section 2.

###  3.1 Equation of state and energy conditions

The simplest and most natural EoS we can choose is

$p_{\parallel}=-\epsilon.$ (9)

This choice is physically well-motivated by the analogy with the cosmological
and galactic regime, since it allows both for a dS and SdS (cosmological)
phases. It allows for a pure dS behavior near $r=0$, which implies the absence
of a singularity in the black-hole interior. Moreover, the EoS also allows for
asymptotically-flat solutions at $r\to\infty$, when both $p_{\parallel}\to 0$
and $\epsilon\to 0$. One can now easily check that, using eq. 5a and eq. 5b,
the EoS (9) implies $\lambda(r)=-\nu(r)$. In the remainder of the paper, we
will adopt the following parametrization of the metric functions
$e^{\nu}=e^{-\lambda}=A(r)$.

Equation 8 can be readily integrated, using eqs. 7 and 9, and yields

$A(r)=1-\frac{2Gm(r)}{r}.$ (10)

Finally, using eqs. 5c, 7 and 9, we can express the fluid anisotropy
$p_{\perp}-p_{\parallel}$ as a function of the MS mass as follows

$\frac{p_{\perp}-p_{\parallel}}{r}=\frac{1}{4\pi
r^{3}}\left(m^{\prime}-\frac{rm^{\prime\prime}}{2}\right).$ (11)

It is useful to write down explicitly the energy conditions for the specific
case in which the EoS (9) holds.

#### Null energy condition (NEC)

In order this condition to be satisfied, we have to require that both
$\epsilon+p_{\parallel}\geq 0$, $\epsilon+p_{\perp}\geq 0$ hold globally
[105]. The first is trivially satisfied due to eq. 9, while the second one
reduces to

$\epsilon^{\prime}(r)\leq 0,$ (12)

upon using eqs. 7 and 11.

#### Strong energy condition (SEC)

In this case, we have to require $\epsilon+p_{\parallel}+2p_{\perp}\geq 0$ to
hold globally [105]. Together with eqs. 7 and 11, this requirement reduces to

$2r\epsilon(r)+r^{2}\epsilon^{\prime}(r)\leq 0.$ (13)

###  3.2 Absence of singularity and behavior near $r=0$

In order to avoid the presence of a central singularity at $r=0$ we first
impose a set of minimal, very general requirements on the form of the metric
functions and on the density and pressure profiles:

  * •

Regularity of $e^{-\lambda}$ in $r=0$ together with eqs. 6 and 10 require
$m(r)\to 0$ sufficiently fast for $r\to 0$;

  * •

Regularity of $p_{\parallel}$ in $r=0$ together with eq. 8 require
$r\nu^{\prime}\to 0$ sufficiently fast for $r\to 0$;

  * •

We also require both $p_{\parallel}$ and $p^{\prime}_{\parallel}$ to be smooth
in $r=0$. From eq. 5c, this implies the regularity condition for the
tangential pressure component

$\lim_{r\to 0}\frac{p_{\perp}-p_{\parallel}}{r}=0.$ (14)

Equation (14), together with eq. 11, implies the following behavior near $r=0$
for the mass function $m(r)$

$m(r)\sim
m_{1}+\frac{r^{3}}{2\ell_{\text{P}}^{2}\hat{L}^{2}}+\mathcal{O}(r^{4})\quad\text{for}\quad
r\to 0$ (15)

with $m_{1}$ and $\hat{L}$ some integration constants. Absence of curvature
singularity for the metric in $r=0$ requires $m_{1}=0$. The other term,
proportional to $r^{3}$, instead, gives a local dS solution with a dS length
$\hat{L}$

$A(r)\sim 1-\frac{r^{2}}{\hat{L}^{2}}+\mathcal{O}(r^{3})\quad\text{for}\quad
r\to 0.$ (16)

This dS description of the solution core is fully consistent with both the EoS
(9) and the analogy with the SdS solution in cosmology (see, e.g. Ref. [8] for
a regular model with a Minkowski core).

###  3.3 Asymptotic flatness and behavior at $r\to\infty$

In the asymptotic ($r\to\infty$) region, our solution must be
indistinguishable from the Schwarzschild solution. This implies the space-time
to be asymptotically flat, with a subleading Schwarzschild term in the metric
function $A(r)$. Moreover, the two pressure components $p_{\parallel}$ and
$p_{\perp}$ have to vanish in the limit $r\to\infty$. Equation 11 implies that
the minimal condition to have $p_{\perp}\to 0$ is

$m=\mathcal{C}_{0}r^{2}+\mathcal{C}_{2}r+M+{\mathcal{O}}(\frac{1}{r}),$ (17)

with $\mathcal{C}_{0},\,\mathcal{C}_{2}$ and $M$ integration constants.
Asymptotic flatness and absence of conical defects require
$\mathcal{C}_{0}=\mathcal{C}_{2}=0$, whereas $M$ becomes the Arnowitt-Deser-
Misner (ADM) mass of the solution measured by the asymptotic observer,

$m=M+{\mathcal{O}}\left(\frac{1}{r}\right)\quad\text{for}\quad r\to\infty.$
(18)

We will also assume that the function $m(r)$ interpolates smoothly between the
near $r=0$ dS behavior and flat space-time at $r\to\infty$.

##  4 A general class of non-singular quantum black-hole models

Absence of a central singularity and the requirement of a Schwarzschild
asymptotic behavior strongly constrain, through eqs. 16 and 18, the local form
of the mass function $m(r)$ (or, equivalently, of the metric function $A(r)$)
near $r=0$ and $r\to\infty$. However, the global behavior of $m(r)$, which
interpolates between $r=0$ and $r\to\infty$, remains extremely weakly
constrained. In this section, we will use the analogy with the cosmological,
SdS case to further constrain its global form. We will get, as output, a
general class of models, which can be used to give an effective description of
quantum black holes.

Using the analogy with the cosmological case, we assume that $m(r)$ depends
only on the two parameters $\hat{L}$ and $R_{\text{S}}$, which characterize
its local behavior near $r=0$ and near $r\to\infty$. This implies that the
quantum IR scale $\ell$ can be written as a function of $\hat{L}$ and
$R_{\text{S}}$ only, $\ell=\ell(\hat{L},R_{\text{S}})$. The explicit relation
between these three scales can be found using a simple argument. As our models
interpolate between the scale-invariant dS behavior in the core and that of
clustered matter, which gives the Schwarzschild solution at large radii, there
will be a scale $\ell$ at which these two effects balance out. By exploiting
the same arguments used in Ref. [58], we expect this scale to correspond to
the one at which the Compton length associated to a test particle, with mass
$m$, in the dS potential $V_{\text{dS}}=r^{2}/\hat{L}^{2}$, is of the same
order of the Compton length of the same test particle in the Schwarzschild
potential generated by the surrounding mass, i.e.
$V_{\text{Sch}}=R_{\text{S}}/r$. The former, curing the singularity at the
center, reads $\lambda_{\text{c,
dS}}\sim\hbar/(|V_{\text{dS}}|m)=\hbar\hat{L}^{2}/r^{2}m$, while the latter,
instead, responsible for quantum correction at the horizon scale, is
$\lambda_{\text{c, Sch}}=\hbar r/R_{\text{S}}m$. We thus have

$\frac{\lambda_{\text{c, dS}}}{\lambda_{\text{c,
Sch}}}\sim\mathcal{O}(1)\Rightarrow r\sim\ell\sim
R_{\text{S}}^{1/3}\hat{L}^{2/3},$ (19)

which is a scaling relation of the form given in eq. 1, with $a=1/3,\,b=2/3$.
We will check this general results in two specific models in section 5 and
section 6.

The presence of a new IR quantum scale implies that we have two complementary
descriptions of the quantum black hole. A black-hole-interior description,
based on the parameters $\hat{L}$ and $M$, and a black-hole-exterior one,
based instead on $\ell$ and $M$. The second case corresponds to the classical
description characterized by the classical hair $M$ and by a quantum
deformation parameter, i.e. the quantum hair $\ell$.

A second requirement on the form of the function $m(r)$ comes from the analogy
with the cosmological SdS space-time case. The space-time must allow for two
horizons, an internal one at $r=r_{-}$ – a dS-like horizon – and an external
one at $r=r_{+}$ – a Schwarzschild-like horizon. Depending on the value of the
parameter $\ell$ (or, equivalently, of the parameter $\hat{L}$), we will have
three cases: $1)$ a black hole with two horizons, $2)$ the two horizons merge
and the black hole becomes extremal, and $3)$ a horizonless compact object.

We can easily estimate the relation between the parameters in the extremal
case $2)$ using a very simple argument. For the internal observer, which
describes its space-time as dS, the energy density is constant and given by
$\epsilon\sim\frac{1}{\hat{L}^{2}\ell_{\text{P}}^{2}}$. The total energy
inside a sphere of radius $r$ is
$E(r)\sim\frac{r^{3}}{\hat{L}^{2}\ell_{\text{P}}^{2}}$. For the extremal black
hole, we must have $r\sim\hat{L}$ and the total energy inside the sphere must
match the black-hole mass $M$ seen by the outside observer:
$E(L)\sim\frac{\hat{L}^{3}}{\hat{L}^{2}\ell_{\text{P}}^{2}}=M$. This equation,
together with eq. 19, implies that the extremal black hole must be
characterized by

$\ell\sim R_{\text{S}}\sim\hat{L}.$ (20)

We find therefore that $\ell$ has both a quantum origin and should be of the
same order of magnitude the Schwarzschild scale $R_{\text{S}}$.

For $\ell\lesssim R_{\text{S}}$, the presence of the dS core and asymptotic
flatness force the metric to have an even number of horizons. In the
following, we will limit ourselves to the case of only two horizons (see
below). In the limit $\ell\to 0$, corresponding to $\hat{L}\to 0$, the inner
dS horizon is pushed towards $r=0$ and disappears: a singularity is generated
in the center. The outer horizon, on the other hand, becomes the classical
Schwarzschild one. This case corresponds to the classical limit of our model,
in which the usual asymptotically-flat Schwarzschild solution is recovered. In
this regime, quantum effects can be neglected. In view of eq. 18, the simplest
way to recover the Schwarzschild solution in the $\ell\to 0$ limit is to
assume that

$m(r)=m\left(\frac{r}{\ell},\,M\right).$ (21)

Conversely, in the $\ell\to\infty$ limit, corresponding also to
$\hat{L}\to\infty$, the outer horizon disappears and the space-time becomes
dS. Notice that also $M\to\infty$ in this case, consistently with the fact
that we have a constant energy density. This is the cosmological regime of
emergent gravity, in which dark energy in the form of the cosmological
constant $L^{-2}$ fully dominates [44]. Our description in terms of a quantum
black hole with a dS interior sourced by an anisotropic fluid breaks down in
this limit. An effective description of gravitational interactions in terms of
GR sourced by an anisotropic fluid is still valid. It can be used to describe
galactic dynamics and the generation of the IR length of galactic size (2),
giving rise to interesting effects, like the emergence of a dark force at
galactic level [57, 56, 58]. Finally, for $\ell\gtrsim R_{\text{S}}$, we have
a solution with no horizons, which can be thought of as an horizonless star.

Let us now write down the most general form of the metric function satisfying
the conditions discussed above and in sections 3.2 and 3.3. We can parametrize
the metric function $A(r)$ in terms of a smooth function $F$ as follows:

$A(r)=1-\frac{R_{\text{S}}}{\ell}F\left(\frac{r}{\ell}\right)\equiv 1-\alpha
F(y)$ (22)

where we have defined the dimensionless coupling $\alpha\equiv
R_{\text{S}}/\ell$ and radial coordinate $y\equiv r/\ell$. Furthermore, the
function $F$ must satisfy the following conditions:

$\displaystyle\quad\,1.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
F(y)\sim\frac{1}{y}\,\quad\text{for}\ \,y\to\infty;$ (23)

$\displaystyle\quad\,2.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
F(y)\sim y^{2}\,\quad\text{for}\ \,y\to 0;$ (24)

  1. 3.

The equation $1-\alpha F(y)=0$ admits at most two real positive roots $y_{+}$,
$y_{-}$. Moreover, parameter regions discriminated by $\alpha$ are present in
which the equation allows for two distinct, a single one or no real positive
roots.

When we have two roots $y_{+}$ corresponds to an event horizon while the inner
horizon, given by $y_{-}$, is generally a Cauchy horizon. The presence of
Cauchy horizons raised several concerns in the literature regarding the
stability and the viability of such regular black-hole solutions with two
horizons. Indeed, according to Poisson, Israel and Ori [106, 107], the Cauchy
horizon is typically exponentially unstable under perturbations, an effect
known as "mass inflation". In standard regular-black-holes approaches with
$\ell\sim\ell_{\text{P}}$, this instability develops in a time of order of the
Planck time, which is a much shorter timescale than the evaporation time,
challenging therefore the ability of these models to give a complete
description of the whole evaporation process [108, 109, 110]. However, it has
been shown recently [111] that a detailed analysis is needed in the non-
singular black-hole case and that the mass instability does not occur in some
specific regular models, like those of Refs. [94, 86] 444We point the reader
to Refs. [112, 113, 114] for an alternative regular model without the presence
of an inner Cauchy horizons in the corpuscular gravity framework. (but see
also Ref. [115] for very recent results).

Apart from the three conditions above on the form of the function $F(y)$, we
also introduce a constraint on the form of its derivative

  1. 4.

In the region $y\geq y_{+}$, $F^{\prime}(y)/F$ has only one local extrema (a
maximum).

The latter represents a statement on the behavior of the black hole
temperature and is needed to have the simplest thermodynamic phase portrait
(see section 4.3).

Finally, requiring the regular quantum-corrected metric to be an exact
solution of some effective field equations derived from an action principle
could place further constraints on the function $F$. These constraints are
analyzed in Ref. [116]. The analysis is mainly concerned with possible quantum
corrections to the Schwarzschild metric, assumed to be polynomial in $1/r$ at
asymptotic infinity (even if the main results of the paper seem to hold even
if the latter assumption is relaxed). These corrections are derived as
asymptotic solutions of effective field equations derived from an Einstein-
Hilbert action corrected by additional higher-order terms in the curvature.
What is found is that, unless either fine-tuning is assumed or strong infrared
non-localities in the gravitational action are taken into account, algebraic
forms of $F$, like those of Refs. [8, 94], for instance, are incompatible with
a principle of least action. Therefore, their feasibility as "quantum-
deformed" black holes may be questionable, at least if one requires these
solutions to be derived from an Einstein-Hilbert action with higher-order
terms in the curvature.

Condition $3$ implies the existence of critical values $\alpha_{c}$ and
$y_{c}$ for $\alpha$ and $y$ respectively, labeling the extremal case, when
the two black-hole horizons merge. These critical values are determined by
requiring the metric function to have a double-degenerate zero, i.e by the
system of equations,

$1-\alpha F(y)=0,\quad F^{\prime}(y)=0.$ (25)

Notice that a simple principle of naturalness implies $\alpha_{c}$ and $y_{c}$
being of order $1$, so that the critical values for $\ell$ and $r$ are of
order $R_{\text{S}}$. Therefore, $\alpha_{c}$ allows us to distinguish three
regimes for our model:

  * $a)$

$\alpha\gg\alpha_{c}$, corresponding to $\ell\ll R_{\text{S}}$ (i.e
$\ell\sim\ell_{\text{P}}$), which describes the Schwarzschild black hole with
small quantum corrections [26, 22, 27, 21, 23, 24, 25, 28, 29, 30, 31, 32];

  * $b)$

$\alpha\sim\alpha_{c}$, corresponding to $\ell\sim R_{\text{S}}$. In this
case, $\ell$ parametrizes quantum gravity effects at horizon scales [41, 42,
43, 44, 45, 46, 47, 48, 49, 50, 54, 55, 74];

  * $c)$

$\alpha<\alpha_{c}$, corresponding to $\ell>R_{\text{S}}$, which describes
horizonless stars. We will not consider this case in the following.

Figure 1: Qualitative behavior of the metric function $A(r)$ as a function of
the radial coordinate and for different values of the parameter $\alpha$. We
can either have solutions with no horizons for $\alpha<\alpha_{c}$ (blue solid
curve), extremal solutions with a single horizon for $\alpha=\alpha_{c}$
(orange dotted curve) or solutions with two horizons for $\alpha>\alpha_{c}$
(green dashed curve).

A qualitative behavior of the metric function $A(r)$ in the three cases is
plotted in Figure 1.

###  4.1 Energy conditions

Given the form (22) for the general metric function and taking into account
(23) and (24), we can rewrite the two energy conditions (12) and (13) in terms
of the function $F$ and discuss their behavior near $r\sim 0$ and for
$r\to\infty$. Using eqs. 6 and 22 in (12) and (13), we get

$\displaystyle\epsilon^{\prime}=-\frac{\alpha F}{4\pi
G\ell^{3}\,y^{3}}+\frac{\alpha F^{\prime\prime}}{8\pi G\ell^{3}\,y}\leq 0;$
(26a) $\displaystyle
2r\epsilon+r^{2}\epsilon^{\prime}=\frac{\alpha\left(2F^{\prime}+yF^{\prime\prime}\right)}{8\pi
G\ell}\leq 0.$ (26b)

Near $y\sim 0$, $\epsilon^{\prime}=0$ follows, so that the NEC is satisfied,
while $2r\epsilon+r^{2}\epsilon^{\prime}\sim 0^{+}$ for $y\to 0$: the SEC is
violated somewhere deep in the core of the astrophysical object. This is
expected, since this violation is a characteristic of the dS spacetime.

On the other hand, for $y\to\infty$, using eq. 23 we get
$\epsilon^{\prime}\sim 0$ and $2r\epsilon+r^{2}\epsilon^{\prime}\sim 0$. Thus,
the NEC and the SEC are satisfied in the asymptotic region, as expected in
view of the Schwarzschild asymptotics of the solutions.

Violation of the SEC in the inner core explains how in our models the usual
singularity theorems can be circumvented. Even if the dS behavior near $r=0$
assures the absence of a curvature singularity at $r=0$, in principle it is
not sufficient to guarantee the geodesic completeness of the space-time
described by the metric function (22). In appendix A we explicitly show that
in our models, caustics of time-like geodesics cannot form, proving therefore
the geodesic completeness of space-times described by eq. 22.

###  4.2 Extremal limit

As it is usually the case for standard charged and/or rotating black holes
[117, 118, 119, 120], in the extremal limit and in the near-horizon
approximation, the local geometry of our space-time behaves as
AdS${}_{2}\times$ S2, i.e as the tensor product of a two-dimensional (2D) AdS
space-time and a two-sphere, with both the AdS2 length $L_{2}$ and the radius
of S2 of order $R_{\text{S}}$. In fact, in the extremal limit
$r_{+}=r_{-}=r_{c}$, the metric function $A(r)$ must have a double zero at
$r=r_{c}$, determined by the solution of eq. 25. Expanding it near the
horizon, i.e. in power series of $r-r_{c}$, we get at leading order

$ds^{2}=-L_{2}^{-2}(r-r_{c})^{2}dt^{2}+L_{2}^{2}(r-r_{c})^{-2}dr^{2}+r_{c}^{2}d\Omega^{2},$
(27)

where we have defined $L_{2}^{-2}=-\frac{1}{2}A^{\prime\prime}(r_{c})$ and
$r_{c}\sim R_{\text{S}}$ owing to eq. 20. Moreover, for purely dimensional
reasons, the same equation implies $A(r_{c})^{\prime\prime}\sim
R_{\text{S}}^{-2}$, from which $L_{2}\sim R_{\text{S}}$ follows. A translation
of the radial coordinate $r\to r+r_{c}$ brings the metric into the form

$ds^{2}=-\frac{r^{2}}{L_{2}^{2}}dt^{2}+\frac{L_{2}^{2}}{r^{2}}dr^{2}+r_{c}^{2}d\Omega^{2},$
(28)

which describes an AdS${}_{2}\times$ S2 space-time, with the AdS2 metric
written in Poincaré coordinates.

As we will see later in this paper, the extremal solution is a zero-
temperature, zero-entropy solution. The extremal configuration will be then
thermodynamically preferred. Even if a solution with two horizons could result
from astrophysical collapse of a compact object, it will decay in a time much
shorter than the Hawking evaporation time into the extremal configuration.

Let us conclude this subsection by noting that the extremal solution is
stabilized by a particular profile for the energy density $\epsilon$ and for
the pressures $p_{\parallel}$ and $p_{\perp}$. In the near-horizon
approximation, when the metric takes the simple AdS${}_{2}\times$S2 form, we
expect them to be constant and to satisfy a simple EoS. In fact, combining
eqs. 5c, 6 and 22, the extremality conditions (25) and the EoS (9), we find
that the leading terms for the near-horizon energy density and pressures are

$\displaystyle\epsilon^{\text{(nh)}}$ $\displaystyle=\frac{1}{8\pi
Gr_{c}^{2}},$ $\displaystyle
p_{\parallel}^{\text{(nh)}}=-\epsilon^{\text{(nh)}},$ (29a)
$\displaystyle\epsilon^{\text{(nh)}}_{\text{(AdS)}}$
$\displaystyle=-\frac{1}{8\pi GL_{2}^{2}},$ $\displaystyle
p^{(\text{nh})}_{\perp}=-\epsilon^{\text{(nh)}}_{\text{(AdS)}},$ (29b)

where $\epsilon^{\text{(nh)}}_{\text{(AdS)}}$ is the (negative) constant
energy density sourcing AdS2. It is quite interesting to note that both the
radial and perpendicular components of the pressure satisfy the simple
equation of state $p=-\epsilon$. The (positive) energy density associated to
the two-sphere acts as a source for the (negative) radial pressure, whereas
the (negative) energy density associated to AdS2 acts as source for the
(positive) perpendicular pressure. Thus, the stabilization of the
AdS${}_{2}\times$S2 near-horizon, extremal solution is achieved in a rather
non-trivial way.

###  4.3 Black hole thermodynamics

From the metric function (22), using standard formulae, we can compute both
the Hawking temperature $T_{\text{H}}$ and the black-hole mass for the quantum
corrected black hole, as functions of the outer horizon radius $r_{+}\equiv
r_{\text{H}}$ and of quantum deformation parameter $\ell$:

$T_{\text{H}}(r_{\text{H}},\ell)=\frac{1}{4\pi}\frac{dA(r)}{dr}\biggr{|}_{r=r_{\text{H}}}=-\frac{\alpha}{4\pi\ell}F^{\prime}\biggr{|}_{y=y_{\text{H}}},\qquad
M(r_{\text{H}},\ell)=\frac{\ell}{2G}F^{-1}(y_{\text{H}}).$ (30)

An important point is that $\ell$ has to be considered as a quantum-
deformation parameter which, contrary to $M$, is not associated with conserved
charges. This makes our quantum black-hole solution drastically different from
other two-parameter classes of solutions, like e.g. the charged RN solution,
for which both parameters are associated with thermodynamic potentials.

Owing to this feature, we look for a first law of thermodynamics of the form
$dM=T_{\text{H}}dS$, where $S$ is the black hole entropy. One can easily check
that the area law, i.e an entropy equal to a quarter of the area of the outer
event horizon (in Planck units), cannot be valid for our class of black-hole
models. In fact, using $S_{A}=\mathcal{A}_{\text{H}}/4G=\pi
r_{\text{H}}^{2}/G$, we get $dM-
T_{\text{H}}dS_{A}=-(1/2G)(dF/dy_{\text{H}})F^{-1}\left(F^{-1}-r_{\text{H}}/\ell\right)dr_{\text{H}}$.
This tells us that, once the area law is assumed, the first principle is
satisfied only for $F(y)=\ell/r_{\text{H}}$, i.e only for the Schwarzschild
black hole.

Let us now look for a new definition of the black hole entropy $S$,
generalizing the area law, such that the first principle is satisfied. This
generalized entropy can be found by noticing that eq. 30 implies the validity
of the following relation

$dM=4\pi MT_{\text{H}}dr_{\text{H}}.$ (31)

By defining the black hole entropy as

$S=4\pi\int Mdr_{\text{H}},$ (32)

we see that the first principle $dM=TdS$ is satisfied. Moreover, the entropy
(32) correctly reproduces the area law in the Schwarzschild case,
$M=r_{\text{H}}/2G$. Equation (32) defines the entropy of the hole up to an
integration constant, which can be fixed by requiring the entropy area law to
be recovered in the limit $\ell\ll R_{\text{S}}$, i.e. in the classical limit
of our quantum model. This leads to

$S(r_{\text{H}})=4\pi\int^{r_{\text{H}}}_{r_{m}}M(r_{\text{H}}^{\prime})dr_{\text{H}}^{\prime},$
(33)

where $r_{m}$ is the minimum value of the horizon radius. In the limit
$\ell\ll R_{\text{S}}$, we have $M(r_{\text{H}})=r_{\text{H}}/2G$ and
$r_{m}=0$, so that eq. 33 gives the area law $S=\pi r_{\text{H}}^{2}/G$.

For a generic quantum deformed black hole, $r_{m}$ is given by the radius
$r_{c}$ of the extremal black hole. This implies in particular that the
extremal black hole has zero entropy, i.e $S(r_{c})=0$. The extremal limit for
our quantum deformed black hole is therefore a state of non-vanishing mass,
but with zero temperature and entropy. Again, this behavior is drastically
different from that of usual extremal black holes, for which the extremal
configuration is a state with $T=0$, but with $S\neq 0$.

Both for large black-hole radii $r_{\text{H}}\to\infty$ and in the extremal
limit, the temperature goes to zero. This can be easily checked using eq. 30
together with eq. 24 and eq. 25. Smoothness of the function $F(y)$ then
implies that the function $T_{\text{H}}(r_{\text{H}})$ has at least one local
maximum in the range $[r_{c}$, $\infty)$. In order to avoid an oscillating
behavior of $T_{\text{H}}(r_{\text{H}})$, we have restricted ourselves to the
simplest case by imposing condition $4$ on the form of the function $F$ (see
the beginning of section 4).

The temperature starts form zero in the extremal limit, then it reaches a
maximum $T_{\text{H, max}}$ for $r_{\text{H, max}}$ and then goes down to zero
again for large values of $r_{\text{H}}/\ell$. This implies that
$T_{\text{H}}$ is always bounded, $0\leq T_{\text{H}}\leq T_{\text{H, max}}$.
Only when we take the limit $\ell\to 0$ first, to recover the Schwarzschild
black hole, can the temperature become arbitrarily large by taking small black
holes, $r_{\text{H}}\to 0$. Notice that a non-vanishing quantum deformation
parameter, $\ell\neq 0$ solves, as expected, the singular thermodynamical
behavior $T_{\text{H}}\to\infty$ of the Schwarzschild black hole for
$r_{\text{H}}\to 0$. The typical qualitative behavior of the temperature is
shown in Figure 2.

Figure 2: Typical qualitative behavior of the black hole temperature
$T_{\text{H}}$ as a function of the black hole radius $r_{\text{H}}$. We
explicitly show the presence of the two branches.

In order to study in detail the thermodynamic behavior of the black hole near
extremality, we expand $T_{\text{H}}$ and $M$ near $r_{c}$. At leading order,
we get $T_{\text{H}}=\gamma(r_{\text{H}}-r_{c})$, whereas
$M=M_{c}+\beta(r_{\text{H}}-r_{c})^{2}$, where
$\gamma=dT_{\text{H}}/dr_{\text{H}}|_{r_{c}}$,
$\beta=(1/2)d^{2}M/dr_{\text{H}}^{2}|_{r_{c}}$ and $M_{c}=M(r_{c})$. Notice
that $dM/dr_{\text{H}}$ is always positive and becomes zero at extremality,
$dM/dr_{\text{H}}|_{r_{c}}=0$. For this reason, the linear term in the
expansion of $M$ is absent. The previous expression implies a quadratic
scaling with the temperature of the mass above extremality

$M-M_{c}\sim\frac{\ell^{3}}{G}T_{\text{H}}^{2},$ (34)

which is fully consistent with the AdS${}_{2}\times$ S2 near-horizon behavior
of the extremal limit [121, 122, 123]. This means that, in the near-extremal
limit, the black hole allows for an effective description in terms of a 2D
dilaton gravity theory, i.e. Jackiw-Teitelboim (JT) gravity, with the dilaton
parametrizing the radius of the 2-sphere S2 [124, 125, 126]. This in turn
implies the possibility of using a dual 2D conformal field theory to describe
the near-horizon regime of the near-extremal black hole. This fact may be
relevant for applications to the information puzzle in black-hole physics [65,
66, 67, 68, 35, 34, 69].

###  4.4 Phase transition

The non-monotonic behavior of $T(r_{\text{H}})$, which is common to a wide
class of charged and/or AdS black holes, signalizes a non-trivial
thermodynamic phase structure, the presence of two thermodynamic branches and
a phase transition at the critical temperature $T_{\text{H, max}}$ (see, e.g.
Refs. [127, 128, 129, 130, 131, 132, 133, 134, 135, 136]). This simply follows
from the fact that we have two possible values of $r_{\text{H}}$ for a given
value of $T_{\text{H}}$. This implies the presence of metastable states and
the existence of two branches, $I$ and $II$. Branch $I$ corresponds to small,
order $1$ values of the dimensionless black hole radius $y_{\text{H}}$ (the
left hand region of Figure 2). In this branch, $r_{\text{H}}$ varies between
the extremal value $r_{c}$ and $r_{\text{H, max}}$. Correspondingly, the
parameter $\ell$ takes values between $\ell_{\text{H, max}}$ and $\ell_{c}$,
both of order $R_{\text{S}}$. Thus, branch $I$ describes quantum black holes
whose quantum deformation parameter $\ell$ is of the same order of magnitude
of the classical Schwarzschild radius $R_{\text{S}}$. Conversely, branch $II$
corresponds to large values of $y_{\text{H}}$ (the right hand region of Figure
2). Here, $r_{\text{H}}$ can take values much larger than $r_{\text{H, max}}$.
This corresponds to small values of the parameter $\ell$. Thus, the far right
region of branch $II$ describes classical black holes, with quantum
deformation parameter $\ell\ll R_{\text{S}}$.

The phase transition and the stability of the different thermodynamic phases
can be investigated by considering the specific heat of the black hole:

$C=\frac{dM}{dT}=\frac{dM}{dr_{\text{H}}}\left(\frac{dT}{dr_{\text{H}}}\right)^{-1}.$
(35)

Being $dM/dr_{\text{H}}$ always positive, the non-monotonic behavior of
$T_{\text{H}}$ implies that

  * •

For $r_{c}<r_{\text{H}}<r_{\text{H, max}}$, $dT/dr_{\text{H}}$ is positive and
thus $C>0$;

  * •

For $r_{\text{H}}>r_{\text{H, max}}$, $dT/dr_{\text{H}}$ is negative and thus
$C<0$;

  * •

For $r_{\text{H}}=r_{\text{H, max}},\,dT/dr_{\text{H}}=0$ and thus
$C\to\infty$.

In Figure 3 we plot the qualitative behavior of the specific heat $C$ as a
function of $r_{\text{H}}$.

Figure 3: Typical qualitative behavior of the black hole specific heat $C$ as
a function of $r_{\text{H}}$. The $r_{\text{H}}$-axis starts from the value at
which $C$ is zero, i.e. at the extremal value of $r_{\text{H}}$, given by the
solution of eq. 25. The vertical dashed line, where $C$ diverges, corresponds
to the zero of $dT/dr_{\text{H}}$, i.e. the maximum of the temperature.

In branch $I$, i.e for values of $r_{\text{H}}$ less than $r_{\text{H, max}}$,
the specific heat is always positive and the black hole exists in equilibrium
with its radiation. On the other hand in branch $II$, i.e for values of
$r_{\text{H}}$ larger than $r_{\text{H, max}}$, the specific heat is always
negative and the black hole cannot be considered at equilibrium with its
radiation anymore. A second order phase transition occurs at $r_{\text{H,
max}}$. This implies that quantum black holes with small values of the quantum
deformation parameter are thermodynamically less favored with respect to black
holes with $\ell\sim R_{\text{S}}$. Moreover, the value of the deformation
parameter $\ell$ at the maximum, $\ell_{\text{H, max}}$, is very close to the
extremal value. This means that black holes close to extremality are
thermodynamically preferred, which further confirms the choice $\ell\sim GM$.
This also shows that the outcome of the evaporation process will be a cold
remnant at zero temperature and zero entropy. The latter, in particular, is
again an intriguing property, as it could allow to circumvent problems on the
viability of remnants as possible solutions on the information paradox [137].

These thermodynamic aspects and the phase picture will be confirmed later on
this paper, when we will consider particular cases of our general class of
models and we will study their free energy.

The non-trivial phase space structure analyzed above is a consequence of the
multiscale description of our models, discussed in section 2. In light of the
similarity between this description of gravitational interactions and glassy
systems [44], one could ask whether our phase transition could be interpreted
as a glass transition. Indeed, even the latter is characterized by a
divergence in the specific heat at the transition point, but it is not
generally classified as a second-order phase transition. To answer this
question, one would need first to define some Ehrenfest equations, to describe
variations of the specific heat and the derivatives of the volume between the
two phases. For a second-order phase transition, both these equations are
satisfied, while either both or one of them is violated in glassy systems. In
the black-hole case, one can define Ehrenfest-like equations [138] by
replacing the volume with the electric charge and/or the angular momentum (if
the model is charged and/or rotating) and analyze their variations between the
two phases.

Here, we can only speculate that the phase transition of our "quantum-
corrected" black hole could be very similar to a glass transition instead of a
second-order one. In fact, the absence of any thermodynamic potentials related
to $\ell$ or other physical observables prevents us from properly define
Ehrenfest-like equations, and therefore to perform an analysis similar to the
one in Ref. [138]. Consequently, this does not allow us to assess
quantitatively the nature of the phase transition in our models.

Summarizing the results obtained so far, the stable configuration of our
quantum-black-hole model, realized using an anisotropic fluid, will be
represented by an extremal (or near-extremal, if we consider small deviations
from extremality) black hole. The black-hole geometry interpolates between a
dS space-time in the black-hole interior (near the singularity), an
AdS${}_{2}\times$ S2 geometry in the near-horizon region, and flat space-time
in the asymptotic, $r\to\infty$, region. The dS behavior near $r=0$ solves the
singularity problem. At extremality, the two (dual) quantum scales
characterizing the quantum black hole ($\ell$, $\hat{L}$) have the same order
of magnitude of the classical black-hole radius $R_{\text{S}}$.

The scale $\ell$ characterizing quantum effects seen by an external observer
is naturally of the order of magnitude of the classical Schwarzschild radius
$R_{\text{S}}$ of the black hole. This opens the possibility of having
phenomenological quantum signatures, potentially observable in the near
future, e.g. through the QNMs spectrum and the geodesic motion of particles
near the horizon. Moreover, the near-horizon AdS${}_{2}\times$ S2 behavior is
very promising for tackling the black-hole information puzzle.

###  4.5 Null geodesics and photon orbits

To compute photon orbits in our class of models, we start from the geodesic
equation together with the null-geodesic constraint (dot will refer to
derivation with respect to the affine parameter)

$\ddot{x}^{\mu}+\Gamma^{\mu}_{\nu\lambda}\dot{x}^{\nu}\dot{x}^{\lambda}=0,\qquad
g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0.$ (36)

The isometries of the metric (spherical symmetry and invariance under time
translations) imply two conservation equations, which by considering geodesics
on the plane $\theta=\text{constant}=\pi/2$, take the form

$\dot{\phi}=\frac{J}{r^{2}},\quad\dot{t}=\frac{\mathcal{C}}{|A(r)|}$ (37)

where $J$ and $\mathcal{C}$ are integration constants.

The geodesic equation for the coordinate $r$ can be integrated to give,

$\dot{r}^{2}+\frac{J^{2}}{r^{2}}A(r)=\mathcal{C}^{2},$ (38)

which can be rewritten as the energy conservation equation:

$\frac{1}{2}\dot{r}^{2}+V(r)=\frac{\mathcal{C}^{2}}{2}\equiv\mathcal{E},$ (39)

with $\mathcal{E}$ a constant and $V(r)$ the effective potential

$V(r)\equiv\frac{J^{2}}{2r^{2}}A(r).$ (40)

Since the leading term of metric function for $r\to 0$ is $A(r)\sim 1$, we
have $V(r)\to\infty$ in this limit. This behavior is completely different from
the Schwarzschild case ($V\to-\infty$ as $r\to 0$) and is a consequence of the
absence of a singularity. Conversely, for $r\to\infty$, $A(r)$ is dominated by
the $1/r$ term and $V(r)\to 0$, as in the Schwarzschild case. The shape of the
effective potential $V(r)$ at intermediate distances depends crucially on the
values of the parameter $\alpha$. The local extrema of $V$ are the solution of
the equation

$rA^{\prime}(r)-2A(r)=2\alpha F-2-\frac{\alpha r}{\ell}F^{\prime}=0.$ (41)

Being $F$ and $F^{\prime}$ both bounded, it will always exist a mimimum value
$\alpha_{m}<\alpha_{c}$ such that for $\alpha\leq\alpha_{m}$ the equation has
no (real) zeroes and bounded photon orbits do not exist. For
$\alpha>\alpha_{m}$, the equation allows instead for two (real) zeroes,
corresponding to a local minimum and a local maximum for $V(r)$. On the other
hand, for $\alpha_{m}<\alpha<\alpha_{c}$, we still have the stable and
unstable photon orbits, but there are no horizons, since the solution
describes a star. For $\alpha=\alpha_{c}$, i.e in the extremal configuration,
the miminum of the potential coincides with the horizon position. In this
case, we will have both the outer unstable photon ring and a stable one,
which, however, coincides with the event horizon of the extremal black hole.
Finally, for $\alpha>\alpha_{c}$, the local minimum is inside the event
horizon and we have only a bounded unstable photon orbit, similarly to the
Schwarzschild case.

The qualitative behavior of $V(r)$ in the four cases is shown in Figure 4.
From the plots we see that there are two main differences with the
Schwarzschild case. The first is the possibility of having a complete absence
of bounded orbits, in the case of the horizonless solution. The second is the
presence of stable orbits for photons in addition to the usual unstable ones.
However, in the two-horizon case, the stable orbits are beyond the black-hole
horizon and do not play any role, while in the extremal case they coincide
with the horizon. Deviations from extremality, however, push this minimum
inside the outer horizon and their effects are irrelevant from the
observational point of view.

(a) $\alpha>\alpha_{c}$ (b) $\alpha=\alpha_{c}$

(c) $\alpha_{m}<\alpha<\alpha_{c}$ (d) $\alpha<\alpha_{m}$

Figure 4: Typical qualitative behavior of the effective potential $V$ for null
geodesics as a function of $r$ for $\alpha>\alpha_{c}$, $\alpha=\alpha_{c}$,
$\alpha_{m}<\alpha<\alpha_{c}$ and $\alpha<\alpha_{m}$.

###  4.6 Quasi-Normal Modes spectrum in the eikonal approximation

In this section we consider QNMs for scalar perturbations in the fixed
background given by our general black-hole solution. We will then use the
eikonal approximation to give an analytical estimate of the quasi-normal
frequencies.

The evolution of scalar perturbations $\Psi$, in the fixed gravitational
background metric $g^{(B)}_{\mu\nu}$ described by the metric function (22), is
determined by the Klein-Gordon equation

$\Box\Psi=\frac{1}{\sqrt{-g^{(B)}}}\partial_{\mu}\left(\sqrt{-g^{(B)}}g_{(B)}^{\mu\nu}\partial_{\nu}\right)\Psi=0.$
(42)

By separating the time, angular and radial parts of $\Psi$, one gets the
radial Schrödinger-like equation for the $r$-dependent part,

$\frac{d^{2}\psi}{dr_{\ast}^{2}}+\left[\omega^{2}-V_{\text{KG}}(r)\right]\psi=0,$
(43)

where the potential is

$V_{\text{KG}}(r)=(1-\alpha
F)\left[\frac{l(l+1)}{r^{2}}-\alpha\frac{F^{\prime}}{\ell r}\right],$ (44)

$l$ is the orbital-angular-momentum number and $r_{\ast}$ is the tortoise
coordinate defined by

$r_{\ast}=\int\frac{dr}{A(r)}.$ (45)

Analytical estimates of the quasi-normal frequencies can be obtained by using
an intriguing relation between quasi-normal modes and the parameters
chracterizing null geodesics in the eikonal limit $l\gg 1$ first noted in Ref.
[139]. The vibration modes of the black holes, whose energy is gradually being
radiated away, are interpreted as photons moving along an unstable null-
geodesics and slowly leaking out. This correspondence was more recently
analyzed in Refs. [140, 141, 142] and shown to agree with WKB results [143,
144]. Specifically, the angular velocity
$\Omega_{m}=\dot{\varphi}/\dot{t}|_{r=r_{m}}$, computed at the maximum of the
geodesic potential $r_{m}$, determines the real part of the quasi-normal
spectrum. Further, the so-called Lyapunov exponent
$\lambda=\sqrt{\frac{V^{\prime\prime}(r)}{2\dot{t}^{2}}}\biggr{|}_{r=r_{m}}$
($V(r)$ is the effective potential for null geodesics (40)), which
characterizes the time-scale of the null-orbit instability, describes the
damping of the black hole oscillations and therefore determines the imaginary
part of the spectrum. We have

$\omega_{\text{QNM}}=\omega_{\text{R}}+i\omega_{\text{I}}=\Omega_{m}l-\text{i}\left(n+\frac{1}{2}\right)|\lambda|,$
(46)

where

$\displaystyle\Omega_{m}=\sqrt{\frac{A^{\prime}(r)}{2r}}\biggl{|}_{r=r_{m}}=\frac{\sqrt{A(r_{m})}}{r_{m}};$
(47a)
$\displaystyle\lambda=\frac{1}{\sqrt{2}}\sqrt{-\frac{r_{m}^{2}}{A(r_{m})}\left(\frac{d^{2}}{dr_{\ast}^{2}}\frac{A(r)}{r^{2}}\right)_{r=r_{m}}}.$
(47b)

Being $r_{m}$ the position of the peak of the geodesic potential, it is given
by the solution of the eq. 41. Although we have seen that, for extremal black
holes, the potential $V$ has an additional minimum exactly at the horizon, we
will not consider this contribution, as small perturbations from extremality
have the effect of moving this minimum towards the black hole interior. The
QNMs spectrum will be therefore entirely determined by the contribution at the
maximum of $V$.

The general expressions (47a) and (47b) are valid for all spherically-
symmetric, static and asymptotically-flat solutions, in the eikonal limit. For
our general class of models, using eqs. 22, 45, 47a and 47b we get

$\displaystyle\omega_{\text{R}}=\frac{l}{r_{m}}\sqrt{A(r_{m})}=\frac{l}{\ell
y_{m}}\sqrt{1-\alpha F_{m}},$ (48a)
$\displaystyle\omega_{\text{I}}=-\left(n+\frac{1}{2}\right)\frac{1}{\sqrt{2}}\sqrt{A(r_{m})r_{m}\biggl{|}\left(\frac{A^{\prime}(r)}{r}\right)^{\prime}_{m}\biggr{|}}=-\left(n+\frac{1}{2}\right)\frac{1}{\sqrt{2}\,\ell}\sqrt{\alpha\left(1-\alpha
F_{m}\right)y_{m}\biggl{|}\left(\frac{F^{\prime}}{y}\right)^{\prime}_{m}\biggr{|}}\,.$
(48b)

It is important to stress that the QNMs frequencies depend both from the
classical hair $M$ and from the quantum hair $\ell$. This dependence from two
parameters of the QNMs spectrum will have a well-defined signature in the
ringdown part of the gravitational wave generated in the merging of two
compact objects to form a single black hole. Next-generation gravitational-
wave detectors are expected to be sensitive enough to detect such effect.

In the generic case, $\omega_{\text{R}}$ and $\omega_{\text{I}}$ will be
complicated functions of $\alpha$. Simpler expressions can be obtained for
near-extremal black holes, by expanding in powers of $(\alpha-\alpha_{c})$.
Taking into account that $|A|_{m}\neq 0$ (see the remark above), if we just
consider the near-extremal expansion but not the near-horizon expansion, we
will also have $\left(\frac{A(r)^{\prime}}{r}\right)^{\prime}_{m}\neq 0$. At
first order in $(\alpha-\alpha_{c})$ we get
$\omega_{\text{I}}=\text{constant}/\ell+\text{constant}\
(G/\ell^{2})(M-M_{c})$ and similarly for $\omega_{\text{R}}$. Using eq. 34, we
can express the quasi-normal frequencies in terms of the black hole
temperature

$\displaystyle\omega_{\text{R}}$ $\displaystyle=\frac{a}{\ell}+b\,\ell\
T_{\text{H}}^{2},$ (49a) $\displaystyle\omega_{\text{I}}$
$\displaystyle=\frac{c}{\ell}+d\,\ell\ T_{\text{H}}^{2},$ (49b)

where $a,b,c,d$ are dimensionless constants.

On the other hand, if we take the near-horizon limit together with the near-
extremal limit, the metric satisfies
$\left(\frac{A(r)^{\prime}}{r}\right)^{\prime}=0$, identically, since the
geometry becomes that of AdS${}_{2}\times$ S2. While the behavior of
$\omega_{\text{R}}$ remains that of eq. 49a, we get a linear scaling of
$\omega_{\text{I}}$ with the temperature, owing to the absence of the constant
term inside the square root:

$\omega_{\text{I}}\propto T_{\text{H}}.$ (50)

These results confirm only partially Hod’s conjecture, which asserts the
complete absence of the imaginary damped part in the spectrum in the near-
extremal case, both for RN and Kerr black holes [75, 76, 77, 78, 79]. In the
case under consideration, Hod’s conjecture seems to hold true only in the
near-extremal, near-horizon case. This seems to be a feature of also general
charged and rotating regular black-hole models [82, 83, 84].

On the other hand, our results seem to confirm a general behavior found in
Ref. [145] for the near-extremal Kerr space-time, which is characterized by a
branching in the quasi-normal spectrum. One family, corresponding to the
simple near-extremal limit, has indeed a non-vanishing imaginary part in the
extremal limit, while the other branch shows that the damped part of the
spectrum goes to zero in the near-extremal, near-horizon limit.

Finally, the temperature scaling (50) fully confirms previous derivation of
the quasi-normal spectrum for two dimensional AdS2 black holes [70, 71, 72,
73, 74]. In fact, the latter allows for an explicit analytical treatment
through different methods, which all point towards the same result: a linear
scaling of the imaginary part $\omega_{\text{I}}$ with the temperature of the
hole. A quite interesting consequence of this scaling is the complete absence
of the imaginary damped part in the spectrum in the extremal case, as the
temperature becomes zero. These zero-damped (or nearly zero-damped) modes [75,
76, 77, 78, 79, 80, 81] would therefore represent a clear phenomenological
signature of extremal black holes.

##  5  A minimal model: the Hayward black hole

The simplest example of our general class of models is given by the Hayward
black-hole metric [94, 146], for which the metric function $F$ in eq. 22 reads

$F(y)=\frac{y^{2}}{y^{3}+1},$ (51)

with $y\geq 0$. As already mentioned in section 4, the analysis of Ref. [116]
shows that an algebraic form of $F$ could be inconsistent with semiclassical
field equations derived from an action principle, at least if if one requires
the solutions to be derived from an Einstein-Hilbert action with higher-order
terms in the curvature.

The horizon location and the extremality condition (25) are now

$y^{3}-\alpha y^{2}+1=0,\quad-2\alpha y+3y^{2}=0.$ (52)

Solving these equations yields the critical values of the parameter $\alpha$
and the critical radius $r_{c}$

$\alpha_{c}=\frac{3}{\sqrt[3]{4}},\quad
r_{c}=\frac{2}{3}R_{\text{S}}=\sqrt[3]{2}\,\ell.$ (53)

The black hole has two horizons for $R_{\text{S}}>3\ell/\sqrt[3]{4}$, is
extremal for $R_{\text{S}}=3\ell/\sqrt[3]{4}$, whereas it becomes a
horizonless star for $R_{\text{S}}<3\ell/\sqrt[3]{4}$. The energy density
$\epsilon$ and the mass function $m$ sourcing the black hole are given by

$\epsilon(r)=\frac{3}{4\pi}\frac{M\ell^{3}}{(r^{3}+\ell^{3})^{2}},\quad
m(r)=\frac{Mr^{3}}{r^{3}+\ell^{3}}.$ (54)

The NEC (12) is always satisfied, while the SEC (13) is violated deep inside
the core of the object, for $r\leq\ell/\sqrt[3]{2}$. On the other hand, for
$\ell\to 0$, or for $r\to\infty$, $\epsilon(r)$ has a very small, Dirac’s
delta-like support only in the region near $r=0$, and therefore $m(r)=M$: the
Schwarzschild black hole is recovered. The deviation from the Schwarzschild
case can be characterized by defining a mass deviation $\Delta m(r)$ as the
difference between the mass at infinity $M$ and $m(r)$, which in the present
case reads

$\Delta m\equiv M-m(r)=\frac{M\ell^{3}}{r^{3}+\ell^{3}}.$ (55)

For $r\to\infty$, it behaves as

$\Delta m\sim\frac{M\ell^{3}}{r^{3}}+\mathcal{O}\left(\frac{1}{r^{4}}\right).$
(56)

The solution is asymptotically flat and satisfies the boundary conditions
$\epsilon$, $p_{\parallel}$, $p_{\perp}\to 0$ for $r\to\infty$, while it has a
dS behavior near $r=0$ with the dS length $\hat{L}$ (15) given by

$\hat{L}=\ell^{3/2}R_{\text{S}}^{-1/2},$ (57)

which relates $\ell$ with the Schwarzschild radius and the dS length
characterizing the small $r$ behavior. Equation 57 fully confirms the validity
of our general scaling given by eq. 19.

We note that the same result in eq. 57 can be obtained in the limit of very
large $\ell$. In this case, however, we have an exact solution of Einstein’s
equations, sourced by a constant-density, isotropic and homogeneous fluid,
with equation of state

$p_{\parallel}=p_{\perp}=-\epsilon=-\frac{3}{8\pi G\hat{L}^{2}}.$ (58)

Indeed, looking at the density profile (51), the dS universe can be recovered
in the limit $\ell\to\infty$ only if $M\to\infty$, so that the energy density
of the source (51) becomes constant.

For $\alpha>\alpha_{c}$, the cubic equation in (53) has three real roots, out
of which two are positive, $r_{+}$ and $r_{-}$, denoting the outer and inner
horizons respectively, whereas the third, $r_{3}$, is negative. The metric
function $A$ factorizes as

$A(r)=\frac{(r-r_{+})(r-r_{-})(r-r_{3})}{r^{3}+\ell^{3}}.$ (59)

In the extremal limit $r_{+}=r_{-}\equiv r_{c}=(2/3)R_{\text{S}}$, after a
translation of the radial coordinate $r\to r+r_{c}$, the space-time metric
becomes

$ds^{2}=-\tilde{A}(r)dt^{2}+\tilde{A}^{-1}(r)dr^{2}+(r+r_{c})^{2}d\Omega^{2},\quad\tilde{A}(r)\equiv\frac{r^{2}(r+r_{c}-r_{3})}{(r+r_{c})^{3}+\ell^{3}},$
(60)

describing an asymptotically flat region connected with an infinitely long
throat of radius $r_{c}$. The near horizon (around $r=0$) expansion of the
metric (60) gives the AdS${}_{2}\times$ S2 space-time (27), with an AdS2
length $L_{2}=(2/3)R_{\text{S}}$. The fluid stabilizing the extremal solution
is characterized by the equations of state (29), where now the AdS2 length
$L_{2}$ and the radius $r_{c}$ of the two-sphere have the same value,
$L_{2}=r_{c}=(2/3)R_{\text{S}}$.

###  5.1 Thermodynamics and phase transition

Inserting $F$ given by eq. 51 into eq. 30, we get the mass and temperature of
the black hole

$T_{\text{H}}=\frac{1}{4\pi\,r_{\text{H}}}\frac{r_{\text{H}}^{3}-2\ell^{3}}{r_{\text{H}}^{3}+\ell^{3}},\quad
M=\frac{1}{2G}\left(r_{\text{H}}+\frac{\ell^{3}}{r_{\text{H}}^{2}}\right).$
(61)

The temperature behavior agrees with the expected qualitative one shown in
Figure 2. The maximum of $T_{\text{H}}$ is obtained by solving the equation
$y^{6}-10y^{3}-2=0$, giving $r_{\text{H}}=\sqrt[3]{(5+3\sqrt{3})}\,\ell$.

Expanding eq. 61 near extremality, one easily finds the quadratic scaling (34)
of the mass with the temperature

$M-M_{c}=12\pi^{2}\frac{\ell^{3}}{G}T_{\text{H}}^{2}.$ (62)

The black-hole entropy satisfying the first principle of thermodynamics is
easily obtained from eq. 32

$S=\frac{\pi r_{\text{H}}^{2}}{G}-\frac{2\pi\ell^{3}}{Gr_{\text{H}}}$ (63)

The first term is the standard area law, while the second term describes
$\ell$-dependent deviations.

Figure 5: Free energy $\mathcal{F}$, in units of $\ell^{-1}$, as a function of
the temperature, in units of $\ell^{-1}$, for the two branches of the Hayward
black hole. We plot $\mathcal{F}$ for branch $I$, $\ell\sim R_{\text{S}}$
(solid blue curve), and for branch $II$ (solid orange curve) corresponding to
$\ell\ll R_{\text{S}}$. We see that "quantum deformed" black holes with
$\ell\sim R_{\text{S}}$ are always energetically preferred with respect to
those with $\ell\ll R_{\text{S}}$.

The specific heat $C$ can easily be calculated using eq. 61 and agrees with
the qualitative behavior shown in Figure 3. It diverges for
$r_{\text{H}}=\sqrt[3]{(5+3\sqrt{3})}\,\ell$, indicating the onset of the
second order phase transition, with the stable thermodynamic branch I
occurring for $r_{\text{H}}<\sqrt[3]{(5+3\sqrt{3})}\,\ell$ 555The possibility
of having a phase transition in the Hayward model has been previously
recognized in Ref. [147].

The existence of this phase transition and related thermodynamical phase
portrait can be checked by computing the free energy
$\mathcal{F}=M-T_{\text{H}}S$ as a function of the temperature. The free
energy for the two branches $I$ and $II$ has to be calculated numerically by
inverting the equation $T_{\text{H}}=T_{\text{H}}(r_{\text{H}})$. We plot
$\mathcal{F}(T_{\text{H}})$ in Figure 5. Branch $I$ corresponds to
$r_{\text{H}}$ varying between the extremal value $r_{c}$ and $r_{\text{H,
max}}$, and thus it describes quantum black holes for which the quantum
deformation parameter $\ell$ is of the same order of magnitude of the
classical Schwarzschild radius $R_{\text{S}}$. Conversely, branch $II$
corresponds to $r_{\text{H}}$ taking values much larger than $r_{\text{H,
max}}$, where $\ell\ll R_{\text{S}}$, corresponding to the classical black-
hole branch.
We note that the branch $I$ is always energetically preferred with respect to
branch II, which further supports our choice $\ell\sim R_{\text{S}}$.

###  5.2 Null geodesics and quasi-normal modes in the eikonal limit

Let us now consider geodesic motion and QNMs for the Hayward black hole. The
effective potential (40), which determines photon orbits in the black-hole
background, in the present case reads

$V(r)=\frac{J^{2}}{2r^{2}}\left(1-\frac{R_{\text{S}}r^{2}}{r^{3}+\ell^{3}}\right).$
(64)

The extrema of $V(r)$ are determined by eq. 41 with $A$ given by eq. 59, i.e.
by the roots of the equation $2y^{6}-3\alpha y^{5}+4y^{3}+2=0$. By solving
this equation numerically, one can show that no real roots exist for
$\alpha<\alpha_{m}\approx 1.64$. The position of the maximum of the potential
will depend on the value of the parameter $\alpha$ with respect to
$\alpha_{m}$ and $\alpha_{c}=3/\sqrt[3]{4}$. Also here, we can distinguish
between the four cases shown in Figure 4. The quasi-normal frequencies in the
eikonal approximation for the Hayward black-hole can be easily calculated
plugging eq. 51 into eqs. 48a and 48b. One has

$\omega_{\text{R}}=\frac{l}{\ell
y_{m}}\sqrt{1-\alpha\frac{y^{2}_{m}}{y^{3}_{m}+1}},\quad\omega_{\text{I}}=-\left(n+\frac{1}{2}\right)\frac{\sqrt{3\alpha}\,y_{m}^{3/2}}{\sqrt{2}\,\ell\left(1+y_{m}\right)^{2}}\sqrt{|y_{m}^{3}-5|\left(1+y_{m}^{3}-\alpha
y_{m}^{2}\right)}.$ (65)

By expanding around $\alpha=\alpha_{c}=3/\sqrt[3]{4}$, we get

$\displaystyle\omega_{\text{R}}\simeq\frac{0.21\,l}{\ell}-\frac{0.11\
l}{\ell}\left(\alpha-\alpha_{c}\right)\simeq\frac{0.21\,l}{\ell}-26.13\,l\
\ell\ T^{2}_{\text{H}};$ (66a)
$\displaystyle\omega_{\text{I}}\simeq-\left(n+\frac{1}{2}\right)\frac{5.70}{\ell}+\left(n+\frac{1}{2}\right)\frac{1.44}{\ell}\left(\alpha-\alpha_{c}\right)\simeq-\left(n+\frac{1}{2}\right)\frac{5.71}{\ell}+\left(n+\frac{1}{2}\right)340.07\
\ell\ T^{2}_{\text{H}},$ (66b)

where in the last equalities we used the definition of $\alpha$ and eq. 62.

##  6 Quantum black holes with gaussian cores

Another explicit and simple realization of a regular quantum black hole,
sourced by an anisotropic fluid with EoS given by eq. 9 and satisfying the
conditions outlined in Section 4 , can be obtained by taking a gaussian
density distribution in the interior of the astrophysical object, peaked at
$r=0$

$\epsilon(r)=\frac{M}{\pi^{3/2}\ell^{3}}e^{-r^{2}/\ell^{2}},$ (67)

where $M$ is the MS mass as seen from infinity, namely

$M=4\pi\int_{0}^{\infty}dr\ r^{2}\epsilon(r).$ (68)

The parameter $\ell$ represents here a smearing of the classical
Schwarzschild-Dirac delta-density profile (the latter is recovered in the
limit $\ell\to 0$). Such a density profile is motivated by several microscopic
descriptions of black holes: non-commutative geometry [22, 148], loop quantum
gravity (LQG) [26], corpuscular picture [149]. The NEC condition (12) is
always satisfied, while the SEC (13) is again violated in the deep core of the
body, i.e. for $r<\ell$.

Near $r=0$, the space-time behaves as dS, with a dS length $\hat{L}$ given by

$\hat{L}=\sqrt{\frac{3}{4}}\pi^{1/4}\,\ell^{3/2}R_{\text{S}}^{-1/2},$ (69)

confirming again our general result given by eq. 19.

As we also saw in section 5, the same result can be obtained as an exact
solution of Einstein’s field equations, sourced by the fluid with EoS (58), in
the limit $\ell\to\infty$. Indeed, looking at the density profile (67), the dS
universe can be recovered in the limit $\ell\to\infty$ only if $M\to\infty$,
and the energy density (67) behaves as a constant.

Let’s turn our attention to the metric structure, by computing the MS mass at
a generic $r$

$\begin{split}m(r)&=4\pi\int_{0}^{r}d\tilde{r}\ \tilde{r}^{2}\
\epsilon(\tilde{r})=\frac{2M}{\sqrt{\pi}}\gamma\left(\frac{3}{2},\frac{r^{2}}{\ell^{2}}\right)=M\left[1-\frac{2}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},\frac{r^{2}}{\ell^{2}}\right)\right]\end{split}$
(70)

where $\gamma(a,z)=\int_{0}^{z}dt\ e^{-t}\,t^{a-1}$ and
$\Gamma(a,z)=\int_{z}^{\infty}dt\ e^{-t}\,t^{a-1}$ are the incomplete gamma
functions. The first term in eq. 70 is the mass measured at infinity (the
Schwarzschild ADM mass), while the second term, parametrized by $\ell$,
encodes the quantum corrections, the effects of the smearing of the
singularity. The deviation from the Schwarzschild solution, described by the
mass deviation $\Delta m(r)\equiv M-m(r)$, is strongly suppressed outside the
core of the compact object, as for $r\to\infty$ it behaves as

$\Delta
m\sim\frac{M}{\sqrt{\pi}}e^{-\frac{r^{2}}{\ell^{2}}}\left(\frac{\ell}{r}+\frac{2r}{\ell}\right)$
(71)

which represents a stronger suppression with respect to that of the Hayward
model (see eq. 56).

The metric components can be written in the form of eq. 22 with the metric
function $F(y)$ given by

$F(y)=\frac{1}{y}\left[1-\frac{2}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},y^{2}\right)\right].$
(72)

Contrary to the Hayward model, in the present case, the non-algebraic form of
$F$ allows to circumvent the viability constraints of Ref. [116].

Given the form of the metric functions, the position of the horizon(s) and the
parameters range discriminating between the two-, one- or no horizons cases
have to be computed numerically. The two horizons merge and we have the
extremal case when both equations in (25) are satisfied. This translates into
the conditions:

$\displaystyle 1-\frac{2GM}{y\ \ell}+\frac{4GM}{\ell\sqrt{\pi}\
y}\Gamma\left(\frac{3}{2},y^{2}\right)=0;$ (73a) $\displaystyle
1-\frac{2}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},y^{2}\right)-\frac{4y^{3}}{\sqrt{\pi}}e^{-y^{2}}=0.$
(73b)

These equations need to be solved numerically 666We discard the simplest
analytical solution, i.e. $r=0$, since it is not a minimum.. The numerical
solution of eq. 73b is $y_{\text{min}}\simeq 1.51$, which means
$r_{\text{min}}\simeq 1.5\,\ell$. The range of parameter
$\alpha=R_{\text{S}}\ell^{-1}$ discriminating between the aforementioned three
cases therefore is

  * •

No horizons: $A\left(y=y_{\text{min}}\right)>0$

$\begin{split}\alpha<\frac{3}{2-\frac{4}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},\frac{9}{4}\right)}.\end{split}$
(74)

  * •

One horizon: $A\left(y=y_{\text{min}}\right)=0$

$\alpha=\frac{3}{2-\frac{4}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},\frac{9}{4}\right)}\simeq
1.90.$ (75)

  * •

Two horizons: $A\left(y=y_{\text{min}}\right)>0$

$\alpha>\frac{3}{2-\frac{4}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},\frac{9}{4}\right)}.$
(76)

In this latter case, the outer horizon is an event horizon, while the inner
one is a Cauchy horizon [148].

We have therefore a critical value $\ell_{c}$ for the quantum parameter $\ell$
for which the two horizons merge and the quantum black hole becomes extremal

$\ell_{c}\simeq 0.53R_{\text{S}},$ (77)

which is close to the classical gravitational radius of the compact object.
This critical value discriminates between the three classes of solutions. When
$\ell$ is above, equal or below $\ell_{c}$ we have a solution with two, one or
none horizons, correspondingly. $\ell_{c}$, in turns, determines a critical
value $\hat{L}_{c}$ for dS length through eq. 69, which also turns out to be
close to the classical Schwarzschild radius 777The fact that
$\hat{L}_{c}<\ell_{c}$ is expected: the SEC is violated in the deep core of
the object, namely for $r<\ell$. This is perfectly consistent with the fact
that this energy condition is violated in a dS space-time.:

$\hat{L}_{c}\simeq 0.31R_{\text{S}}.$ (78)

The most interesting case is the extremal black hole, obtained for
$\ell=\ell_{c}$ ($\hat{L}=\hat{L}_{c}$). As discussed in sect. 4, in the
extremal case the near-horizon geometry factorizes as AdS${}_{2}\times$ S2,
with the AdS2 length given by
$L_{2}^{-1}=\sqrt{-\frac{1}{2}A(r_{c})^{\prime\prime}}$. $L_{2}$ can be
calculated first using eqs. 73a and 73b to get

$e^{-y^{2}(r_{c})}=\frac{\sqrt{\pi}\ell}{4R_{\text{S}}y^{2}(r_{c})}.$ (79)

Using this equation together with eqs. 73a, 73b, 70 and 72 enables us to find

$-\frac{1}{2}A(r_{c})^{\prime\prime}\equiv\frac{1}{L_{2}^{2}}=\frac{1}{\ell^{2}}-\frac{1}{r_{c}^{2}}.$
(80)

Using eq. 77 and taking into account that $r_{\text{min}}\simeq
1.5\,\ell\simeq 0.78\,R_{\text{S}}$ we get

$L_{2}\simeq 0.70\,R_{\text{S}},$ (81)

confirming the general result of Sect. 4, according to which both the radius
of the two-sphere and the AdS2 length are of order $R_{\text{S}}$.

As shown at the end of section 4.2, the extremal solution is stabilized by a
particular profile for the energy density $\epsilon$ and for the pressures
$p_{\parallel}$ and $p_{\perp}$, given by the expressions (29). In the present
case, the negative energy density sourcing the AdS2 space-time reads
$\epsilon^{\text{(nh)}}_{\text{(AdS)}}\equiv-1/(8\pi GL_{2}^{2})=-(1/8\pi
G)\left(1/\ell^{2}-1/r_{c}^{2}\right)$.

###  6.1 Quantum black hole regimes

The parameter $\ell$ (or equivalently $\hat{L}$) controls the scale of quantum
effects in our quantum black-hole model. In the usual, most conservative
approach, which assumes quantum gravity effects to be relevant only at the
Planck scale $\ell_{\text{P}}$, $\ell$ is assumed to be of the same order of
magnitude of $\ell_{\text{P}}$. This assumption is surely justified when
$\ell$ has an explicit origin in the microscopic description of gravity at the
Planck scale. It is for instance the case of Refs. [22, 148, 149, 26], where
the gaussian model, and the parameter $\ell$ in particular, parametrizes UV
non-commutative [22] or LQG [26] effects.

However, this is not the case in those approaches, like the one followed in
this paper, in which an IR quantum scale, hierarchically smaller than
$\ell_{\text{P}}$, is generated. Given the attention the model with
$\ell\sim\ell_{\text{P}}$ received in the past [26, 22, 27, 21, 23, 24, 25,
28, 29, 31], it is worthwhile to quantitatively compare the two different
regimes $1)$ $\ell\sim\ell_{\text{P}}$ and $2)$ $\ell\sim R_{\text{S}}$ for
the black-hole model with a gaussian core, and analyze the possible impact on
observable phenomenology in the two cases. Again, we remind that we are
considering macroscopic black holes, i.e $R_{\text{S}}\gg\ell_{\text{P}}$.

#### $\ell\sim\ell_{\text{P}}$

In this case, eq. 76 tells us that we always have two horizons, the black hole
is far from extremality and the inner horizon is very close to $r=0$. It is
quite easy to understand that these quantum effects are completely irrelevant
for macroscopic black holes, at least for what concerns the phenomenology
accessible to external observers. In fact, the matter density is sensibly
different from zero only at distances of order $\ell_{\text{P}}$ from the
classical singularity at $r=0$. Therefore, for the external observer, the
deviations from the Schwarzschild solution are expected to be incredibly
small. We have sensible deviations from $r_{\text{H}}\sim R_{\text{S}}$ only
for black holes with masses of order the Planck mass
$m_{\text{P}}=1/\sqrt{G}$, where the event horizon is slightly less than
$R_{\text{S}}$ (the radius of the outer horizon is $r_{\text{H}}\simeq
1.8\sqrt{G}=0.9R_{\text{S}}$). However, as the mass increases, the outer
horizon becomes rapidly indistinguishable from $R_{\text{S}}$. For example,
for a solar mass black hole, $M=1\ M_{\odot}$, the outer horizon of the metric
(72) is at $R_{\text{S}}$ and the corrections are exponentially suppressed by
a factor
$\Gamma\left(\frac{3}{2},\frac{r_{\text{H}}^{2}}{\ell_{\text{P}}^{2}}\right)\sim
e^{-r_{\text{H}}^{2}/\ell_{\text{P}}^{2}}=e^{-R_{\text{S}}^{2}/\ell_{\text{P}}^{2}}=e^{-\frac{4GM^{2}_{\odot}}{c\hbar}}\sim
e^{-10^{76}}$ (note that, in the last expression, we have reinstated the speed
of light $c$ and the Planck constant $\hbar$). The mass deviation at the
horizon $\Delta m(r_{\text{H}})$ is of the same order of magnitude, which is
effectively zero from the point of view of the external observer.

#### $\ell\sim R_{\text{S}}$

As explained in the introduction, there are several indications pointing at
the relevance of quantum effects at horizon scales. However, presently we do
not have a precise microscopic description of these quantum effects, but only
some quite interesting proposals, like e.g. fuzzballs, firewalls, non-local
effects and corpuscular models. An interesting explicit corpuscular black-hole
model with a gaussian core is the one given in Ref. [149], whose density
profile is

$\epsilon_{\text{corpuscular}}=\frac{7^{2}m_{\text{P}}}{\sqrt{\pi}N}\frac{e^{-\frac{7}{2}\frac{r^{2}}{N\ell_{\text{P}}^{2}}}}{\ell_{\text{P}}^{3}}$
(82)

where $N$ is the number of gravitons building up the black hole. Comparing
this profile with our model (67), we can read the values of our parameters
$\ell$, $M$ in terms of $N$

$\ell=\ell_{\text{P}}\sqrt{\frac{2N}{7}},\qquad
M=49\pi\frac{\ell^{3}}{N\ell_{\text{P}}^{4}}.$ (83)

From these equations, one easily gets the expected holographic scaling of $N$,
$N\propto\ell_{\text{P}}^{2}M^{2}$ and a value of $\ell$ which is
$\ell=R_{\text{S}}/28\pi\simeq 0.01R_{\text{S}}<\ell_{c}$. The black hole has
two horizons and is far from extremality. The outer horizon is quite close to
the classical Schwarzschild one, we have $r_{\text{H}}\simeq
0.96R_{\text{S}}$. Therefore, the mass deviation is again quite small:

$\frac{\Delta
m}{M}\biggr{|}_{\text{H}}=\frac{2}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},y_{\text{H}}^{2}\right)\sim
e^{-7733}.$ (84)

We see that, for a value of $\ell$ which is about $1/100$ of the critical
value $\ell_{c}$, deviations from the classical behavior are still quite
small.

As a last example we consider a value $\ell<\ell_{c}$, but quite close to the
critical value, $\ell=\ell_{c}/2$. The space-time has two horizons and the
outer one is at

$r_{\text{H}}\simeq 0.92R_{\text{S}},$ (85)

which is a small, but still important, difference with respect to the
classical radius $R_{\text{S}}$. The mass deviation is

$\frac{\Delta
m}{M}\biggr{|}_{\text{H}}=\frac{2}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},\frac{c^{4}r^{2}_{\text{H}}}{G^{2}M^{2}}\right)\simeq\frac{2}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},1.84^{2}\right)\simeq
0.07.$ (86)

These results further show that the most interesting regime is that for which
$\ell\sim R_{\text{S}}$ from both a purely theoretical and from a
phenomenological points of view.

###  6.2 Thermodynamics and phase transition

Figure 6: Plot of the entropy of the quantum gaussian black hole (dashed
orange line) vs standard Hawking entropy of the Schwarzschild black hole
(minus the corresponding Hawking entropy of the extremal case). For
simplicity, we set $\ell=1$ and $G=1$.

Inserting the metric function (72) into eqs. (30), we get the temperature and
the ADM mass of the gaussian black holes

$T_{\text{H}}=\frac{1}{4\pi
r_{\text{H}}}\left[1-\frac{8GM(r_{\text{H}})r^{2}_{\text{H}}}{\ell^{3}}e^{-r^{2}_{\text{H}}/\ell^{2}}\right],\quad
M(r_{\text{H}})=\frac{r_{\text{H}}}{2G\left[1-\frac{2}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},\frac{r_{\text{H}}^{2}}{\ell^{2}}\right)\right]}.$
(87)

The temperature is given by the standard Hawking result plus an
$\ell$-dependent term, encoding quantum deviations from standard black hole
thermodynamics, which however are exponentially suppressed. The behavior
agrees with the qualitative one depicted in Figure 2. It starts from zero in
correspondence with the extremal case. Then, it rises and reaches a maximum,
whose position is given by solving the equation
$dT_{\text{H}}/dr_{\text{H}}=0$, which in the present case is at $r_{\text{H,
max}}\simeq 2.38\ell$. Finally, it decreases and reaches zero as
$r_{\text{H}}/\ell\to\infty$, in agreement with the fact that we have to
recover the standard Schwarzschild results $T_{\text{H}}=\frac{1}{4\pi
r_{\text{H}}},\,\,M=\frac{r_{\text{H}}}{2G}$ in this limit. Also in this case,
the quantum deformation parameter $\ell\neq 0$ solves the singular
thermodynamic behavior of the Schwarzschild temperature
$T_{\text{H}}\to\infty$ for $r_{\text{H}}\to 0$.

On the other hand, from eq. 73a, we get the value of the deformation parameter
corresponding to $r_{\text{H, max}}$

$\ell_{\text{H,
max}}=\frac{1-\frac{2}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},y^{2}_{\text{H,
max}}\right)}{y_{\text{H, max}}}R_{\text{S}}\simeq 0.42R_{\text{S}}.$ (88)

The entropy of the black hole can be obtained by integrating eq. 33
numerically, using the fact that the zero-entropy state is at the extremal
radius $r_{\text{H, extremal}}/\ell\simeq 1.51$. The result of the integral is
presented in Figure 6 (dashed orange line) and is compared to the standard
result for the Schwarzschild black hole 888In this case, the blue solid curve
in Figure 6 is obtained by subtracting the Hawking entropy of the extremal
configuration $S_{\text{extremal}}=\pi r^{2}_{\text{H,extremal}}$ (with $G=1$)
from the standard Schwarzschild entropy $\pi r_{\text{H}}^{2}$.. As it can be
seen, the entropy does not differ significantly from the standard area law, as
quantum deviations are expected to be exponentially suppressed [150, 151, 152]
(see also eq. 71).

Expanding eq. 87 near extremality yields the quadratic scaling (34) of the
mass above extremality with the temperature

$M-M_{c}\simeq 15.55\frac{\ell^{3}}{G}T_{\text{H}}^{2}.$ (89)

The specific heat can be computed using eq. 35 and follows the qualitative
behavior of Figure 3: it diverges at $r_{\text{H, max}}\simeq 2.38\,\ell$,
indicating the onset of the second-order phase transition. Indeed, by
computing numerically the free energy $\mathcal{F}=M-T_{\text{H}}S$ and
expressing it as a function of $T_{\text{H}}$, we get the phase diagram
depicted in Figure 7.

Figure 7: Free energy $\mathcal{F}$, in units of $\ell^{-1}$, as a function of
the temperature in units of $\ell^{-1}$, for the gaussian model. We
distinguish between two branches: one corresponding to black holes with
$\ell\sim R_{\text{S}}$ (solid blue curve), the other (orange curve)
corresponding to $\ell\ll R_{\text{S}}$.

Again, we have two branches. Branch $I$ corresponds to $r_{\text{H}}$ taking
values between the extremal value $r_{c}$ and $r_{\text{H, max}}$
(correspondingly, the parameter $\ell$ takes values between $\ell_{\text{H,
max}}\simeq 0.42R_{\text{S}}$ and $\ell_{c}\simeq 0.53R_{\text{S}}$), and
therefore describes quantum black holes with $\ell\sim R_{\text{S}}$.
Conversely, branch $II$ corresponds to $r_{\text{H}}$ much larger than
$r_{\text{H, max}}$, corresponding to small ($<0.42R_{\text{S}}$) values for
the parameter $\ell$. Thus, the far right region of branch $II$ describes
again classical black holes ($\ell\ll R_{\text{S}}$).

Figure 7 shows that the free energy of branch $I$ is always smaller than that
of branch $II$. This means that black holes in branch $I$, i.e. black holes
with $\ell\sim R_{\text{S}}$, are always energetically preferred.

###  6.3 Null geodesics and photon orbits

The effective potential (40) determining photon orbits in the gaussian quantum
black hole reads

$V(r)=\frac{J^{2}}{2r^{2}}\left[1-\frac{2GM}{r}+\frac{4GM}{\sqrt{\pi}r}\Gamma\left(\frac{3}{2},\frac{r^{2}}{\ell^{2}}\right)\right],$
(90)

and its behavior for different values of the parameter $\alpha\equiv
R_{\text{S}}\ell^{-1}$ agrees with the qualitative plots shown in Figure 4.

The extrema of the potential are given by the zeros of $dV(r)/dr$ (see eq.
41), i.e. by the solution of the equation

$\begin{split}&-r+3GM-\frac{4GM}{\sqrt{\pi}}\frac{r^{3}}{\ell^{3}}e^{-r^{2}/\ell^{2}}-\frac{6GM}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},\frac{r^{2}}{\ell^{2}}\right)=0.\end{split}$
(91)

We see that the presence of $\ell$ shifts the position of the photon sphere
from the Schwarzschild case $r=3GM$.
By numerically solving the equation above, we distinguish again between the
three cases (2, 1 or no horizons)

  * •

Two horizons
($\alpha>\frac{3}{2-\frac{4}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},\frac{9}{4}\right)}$):
we have multiple zeros, corresponding to one minimum in the black-hole
interior and one maximum in the exterior, which corresponds to the position of
the unstable photon orbit. We are interested in the latter. For instance, for
$\alpha=2$ ($\ell=GM$ therefore), we have $y_{m}\simeq 2.99$, which
corresponds to a position of the photon ring $r_{m}\simeq 2.99\ell=2.99GM$,
very close to the Schwarzschild case $3GM$.

  * •

One horizon
($\alpha=\frac{3}{2-\frac{4}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},\frac{9}{4}\right)}$):
this case is similar to the previous one, we have a minimum (coinciding with
the event horizon), which represents a stable photon orbit, and a maximum in
the potential, representing the unstable photon orbit. Focusing again on the
latter, we have $y_{m}\simeq 2.84$, corresponding to $r_{m}\simeq 2.98GM$,
again pretty close to the Schwarzschild result.

  * •

No horizons
($\alpha<\frac{3}{2-\frac{4}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},\frac{9}{4}\right)}$):
we have two possible cases. If $\alpha$ is large enough
($\alpha\sim[1.6,1.8]$), we can have a maximum corresponding to a photon ring.
For example, for $\alpha=1.6$, we get $y_{m}\simeq 2.19$, which means
$r_{m}\simeq 2.74GM$, which is a significant departure from the standard
result. However, if $\alpha$ falls below the aforementioned interval, we do
not have a maximum anymore and the photon ring cannot be present.

###  6.4 Quasi-normal modes spectrum in the eikonal limit

We can use the general arguments and results of Section 4.6 to compute the
expression of the quasi-normal frequencies for the quantum gaussian black hole
in the eikonal limit. Applying the general formulae (48a), (48b) to the case
of eq. 72, we get

$\displaystyle\omega_{\text{R}}=\frac{l}{\ell
y_{m}}\sqrt{1-\frac{\alpha}{y_{m}}\left[1-\frac{2}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},y_{m}^{2}\right)\right]}$
(92a)
$\displaystyle\omega_{\text{I}}=-\left(n+\frac{1}{2}\right)\frac{1}{\sqrt{2}\ell}\sqrt{\alpha\left[y_{m}-\alpha+\frac{2}{\sqrt{\pi}}\Gamma\left(\frac{3}{2},y_{m}^{2}\right)\right]\biggl{|}\frac{3}{y_{m}^{4}}-\frac{4}{\sqrt{\pi}}\frac{1+2y_{m}^{2}}{y_{m}}e^{-y^{2}_{m}}-\frac{6}{\sqrt{\pi}y^{2}_{m}}\Gamma\left(\frac{3}{2},y_{m}^{2}\right)\biggr{|}}$
(92b)

By expanding around the critical value $\alpha_{c}$ (75), we get

$\displaystyle\omega_{\text{R}}\simeq\frac{l}{\ell}\left[0.20-0.10(\alpha-\alpha_{c})\right]\simeq
0.20\frac{l}{\ell}-3.36\,l\ \ell\ T_{\text{H}}^{2};$ (93a)
$\displaystyle\omega_{\text{I}}\simeq\frac{1}{\ell}\left(n+\frac{1}{2}\right)\left[-0.19+0.05(\alpha-\alpha_{c})\right]\simeq-\frac{0.19}{\ell}\left(n+\frac{1}{2}\right)+1.63\left(n+\frac{1}{2}\right)\ell\
T_{\text{H}}^{2}.$ (93b)

##  7 Conclusions

In this paper, we explored the possibility of relating long-range quantum
gravity corrections at horizon scales with the absence of the central
singularity in the Schwarzschild black hole. In order to achieve this, we
explicitly constructed a general class of static, non-singular,
asymptotically-flat black-hole models as exact solutions of Einstein’s field
equations sourced by an anisotropic fluid with EoS
$p_{\parallel}=-\varepsilon$. A natural physical consequence of this choice is
the fact that these models possess a dS core. This causes a violation of the
strong energy condition near the black-hole center, which therefore allows to
circumvent Penrose’s singularity theorem.

Analogously to what has been done at galactic and cosmological level [57, 56,
58, 59, 60, 61], the anisotropic-fluid source is used to give a multi-scale
description of the gravitational system and an effective description of the
quantum black hole. In our models, quantum corrections are effectively encoded
in a single additional scale $\ell$, which appears as an extra "quantum hair"
for the black hole and is related to the dS length $\hat{L}$, smearing the
would-be central singularity of the Schwarzschild black hole.

Apart from solving the singularity problem and allowing for quantum effects at
horizon scales, our general black-hole solutions have also other rather
interesting geometric and thermodynamic features. They allow for the presence
of two horizons, an outer event horizon and an inner Cauchy one. When these
two merge in a single horizon, we have an extremal configuration, which is a
zero-temperature, zero-entropy state, whose near-horizon geometry factors as
AdS${}_{2}\times$ S2. This fact could be very useful in addressing the black-
hole information puzzle.

The previous features determine a rather non-trivial thermodynamic behavior.
The presence of the quantum hair $\ell$, not related to a thermodynamic
potential, required a generalization of the area law for the black-hole
entropy. We also found a metastable behavior and a phase transition, in which
black holes closer to extremality, i.e. with $\ell\sim R_{\text{S}}$, are
always thermodynamically preferred with respect to those with $\ell\gg
R_{\text{S}}$. This means that our "quantum black holes" are thermodynamically
preferred with respect to those in which quantum corrections are irrelevant.
This provided further evidence for the relevance of quantum effects at horizon
scales.

For $\ell\gtrsim R_{\text{S}}$, instead, we found that our models represent
horizonless compact object, which however were not investigated in depth in
the present paper and will be matter for further investigations.

Finally, we have also shown that, when $\ell\ll R_{\text{S}}$, the
phenomenology of our non-singular black-hole solutions is almost
indistinguishable from the standard Schwarzschild-solution one for an outside
observer. On the other hand, for $\ell\sim R_{\text{S}}$, there could be
manifest signatures of deviations from the standard behavior, both in the
photon orbits and in the QNMs spectrum.

For what concerns photon orbits, we have found stable bounded orbits, which
are, however, internal to (in the two-horizon model) or coinciding with (in
the extremal case) the horizon. An interesting result is that for some
horizonless models, unstable photon orbits do not form.

On the QNMs side, we have investigated the spectrum of quasi-normal
frequencies in the eikonal limit and in the near-extremal limit. We found a
scaling behavior with the temperature and a dependence from the quantum hair
$\ell$. These features have a manifest experimental signature, which could be
experimentally accessible in the near future by the next generation of GW
detectors. Taking a near-horizon limit together with a near-extremal one,
instead, we found an imaginary part which vanishes with the temperature. Our
findings only partially confirm Hod’s conjecture on zero-damped QNMs for
standard extremal black holes [75, 76, 77, 78, 79, 80, 81]. On the other hand,
this result is perfectly consistent with the AdS${}_{2}\times$ S2 behavior of
the near-horizon, near-extremal metric and with what is known about the QNMs
spectrum of 2D AdS gravity models [70, 71, 72, 73, 74].

We have confirmed our general results by thoroughly investigating two
particular cases, which represent two widely-known non-singular black-hole
models, namely the Hayward and the gaussian-core metrics. We have revisited
these models considering the quantum deformation parameter of the same order
of magnitude of the Schwarzschild radius $R_{\text{S}}$. This perspective is
completely different from the standard approach to these models, where quantum
effects, and hence $\ell$, are of the order of magnitude of Planck length,
implying extremely small and phenomenologically irrelevant deviations from
standard behavior.

## Acknowledgements

We thank S. Liberati for useful discussions.

##  Appendix A Geodesic completness of the spacetime

In this appendix we show that space-times described by eq. 22 are geodesically
complete. We start from Raychaudhuri’s equation, which describes the evolution
of a time-like geodesic congruence $\Theta$

$\frac{d\Theta}{d\tau}=-\frac{1}{3}\Theta^{2}-\sigma^{\mu\nu}\sigma_{\mu\nu}+\omega^{\mu\nu}\omega_{\mu\nu}-R_{\mu\nu}u^{\mu}u^{\nu},$
(94)

where $\tau$ is the proper time, $u^{\mu}=dx^{\mu}/d\tau$ the proper time-like
velocity, while $\sigma_{\mu\nu}=\Theta_{\mu\nu}-\frac{1}{3}\Theta h_{\mu\nu}$
is the shear tensor ($h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$ is the transverse
metric) and
$\omega_{\mu\nu}=h^{\alpha}_{\mu}h^{\beta}_{\nu}u_{[\alpha;\beta]}$ is the
vorticity tensor. If we consider geodesics to be hypersurface orthogonal, then
$\omega_{\mu\nu}=0$. Since both the shear and the vorticity tensors are purely
spatial, i.e. $\omega_{\mu\nu}\omega^{\mu\nu}\geq 0$,
$\sigma_{\mu\nu}\sigma^{\mu\nu}\geq 0$, and if we assume the strong energy
condition to hold, i.e. $R_{\mu\nu}u^{\mu}u^{\nu}\geq 0$, we expect in all
generality a focusing of the geodesic congruence, i.e. $d\Theta/d\tau\leq 0$
and the formation of caustics, which represent singularities of the
congruence. This is the essence of the original Penrose singularity theorem
[1]. Let us now show that caustics cannot form for the models described by eq.
22. We consider first a generic metric function $A$ and time-like geodesics

$g_{\mu\nu}u^{\mu}u^{\nu}=-1,$ (95)

and focus on geodesics on the plane $\theta=\text{constant}$ and
$\varphi=\text{constant}$. We then have $u^{\theta}=u^{\varphi}=0$. We are
left with

$g_{\mu\nu}u^{\mu}u^{\nu}=-1\Rightarrow-A(r)(u^{0})^{2}+\frac{(u^{r})^{2}}{A(r)}=-1.$
(96)

But $u^{0}=dt/d\tau=1/A(r)$ (since the metric redshift factor determines the
relation between the coordinate and proper time), and therefore

$-\frac{1}{A(r)}+\frac{(u^{r})^{2}}{A(r)}=-1\Rightarrow
u^{r}=\pm\sqrt{1-A(r)}.$ (97)

This yields the congruence

$\Theta=\frac{1}{r^{2}}\partial_{r}\left(r^{2}u^{r}\right)=\frac{1}{r^{2}}\left[2ru^{r}+r^{2}\frac{du^{r}}{dr}\right]=\pm\left(\frac{2}{r}\sqrt{1-A(r)}-\frac{A^{\prime}(r)}{2\sqrt{1-A(r)}}\right).$
(98)

The geodesic congruence evolution as a function of the proper time can
therefore be written as

$\frac{d\Theta}{d\tau}=\frac{d\Theta}{dr}\frac{dr}{d\tau}=\Theta^{\prime}u^{r}=-\frac{2(1-A(r))}{r^{2}}-\frac{A^{\prime}(r)}{r}-\frac{A^{\prime\prime}(r)}{2}-\frac{A^{\prime}(r)^{2}}{4(1-A(r))},$
(99)

where we used the fact that $\Theta$ is a function of $r$ only.

Let us now specialize eq. 99 to the case given by eq. 22. We get

$\frac{d\Theta}{d\tau}=-\frac{2\alpha
F(y)}{y^{2}\ell^{2}}+\frac{\alpha}{y\ell^{2}}F^{\prime}(y)+\frac{\alpha}{2\ell^{2}}F^{\prime\prime}(y)-\frac{\alpha}{4\ell^{2}}\frac{F^{\prime}(y)^{2}}{F(y)}$
(100)

Since, for large $y$, our general model behaves essentially as the
Schwarzschild black hole, we focus on the behavior of the geodesics bundle in
the core of the object, i.e. for $y\to 0$, where $F\sim y^{2}$ according to
eq. 24. Therefore, $F^{\prime}\sim 2y$ and $F^{\prime\prime}\sim 2$, and thus

$\frac{d\theta}{d\tau}\sim-\frac{2\alpha}{\ell^{2}}+\frac{2\alpha}{\ell^{2}}+\frac{\alpha}{\ell^{2}}-\frac{\alpha}{\ell^{2}}\sim
0,$ (101)

so that we do not encounter a caustic in the center, the bundle of geodesics
is defocused and therefore they can be extended beyond $r=0$. This is
consistent with the form of the Penrose diagram for such models (see e.g.
Refs. [148, 94]), which shows that, apart from the presence of the central
singularity, the maximal extension of these space-times is similar to that of
RN black holes.

These results can be further confirmed by computing $d\Theta/d\tau$ for the
two specific models investigated in detail in the paper, namely the Hayward
and the gaussian-core black holes.

For the Hayward black hole, plugging the function $F$ (51) into eq. 100 yields

$\frac{d\Theta}{d\tau}=-\frac{9\alpha
y^{3}\left(y^{3}+4\right)}{4\ell^{2}\left(1+y^{3}\right)^{3}}.$ (102)

Near $y\sim 0$, we have $\frac{d\Theta}{d\tau}\sim-\frac{9\alpha
y^{3}}{\ell^{2}}+\mathcal{O}(y^{4})$, so no caustic is present in the
interior, at $r=0$.

In the gaussian-core black-hole case, using the function $F$ (72) into eq.
100, we get

$\frac{d\Theta}{d\tau}=-\frac{\alpha
e^{-2y^{2}}}{4\ell^{2}\sqrt{\pi}y^{3}}\cdot\frac{9\pi
e^{2y^{2}}+16y^{6}+8\sqrt{\pi}e^{y^{2}}y^{3}\left(2y^{2}-3\right)-4e^{y^{2}}\left(9\sqrt{\pi}e^{y^{2}}-12y^{3}+8y^{5}\right)\Gamma\left(\frac{3}{2},y^{2}\right)+36e^{2y^{2}}\Gamma^{2}\left(\frac{3}{2},y^{2}\right)}{\sqrt{\pi}-2\Gamma\left(\frac{3}{2},y^{2}\right)}.$
(103)

Near $y\sim 0$, we have $\frac{d\Theta}{d\tau}\sim-\frac{4\alpha
y^{2}}{\sqrt{\pi}\ell^{2}}+\mathcal{O}(y^{4})$, so that again caustics do not
form and the space-time is geodesically complete.

## References

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