11institutetext: Alfio Bonanno 22institutetext: INAF, Osservatorio Astrofisico
di Catania, via S.Sofia 78, I-95123 Catania, Italy;
INFN, Sezione di Catania, via S.Sofia 64, I-95123, Catania, Italy
22email<EMAIL_ADDRESS>33institutetext: Frank Saueressig
44institutetext: Institute for Mathematics, Astrophysics and Particle Physics
(IMAPP), Radboud University,
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
44email<EMAIL_ADDRESS>
# Stability properties of Regular Black Holes

Alfio Bonanno and Frank Saueressig

###### Abstract

Black holes encountered in general relativity are characterized by spacetime
singularities hidden within an event horizon. These singularities provide a
key motivation to go beyond general relativity and look for regular black
holes where the spacetime curvature remains bounded everywhere. A prominent
mechanism achieving this replaces the singularity by a regular patch of de
Sitter space. The resulting regular geometries exhibit two horizons: the outer
event horizon is supplemented by an inner Cauchy horizon. The latter could
render the geometry unstable against perturbations through the so-called mass-
inflation effect, i.e., an exponential growth of the mass function. This
chapter reviews the mass-inflation effect for spherically symmetric black hole
spacetimes contrasting the dynamics of the mass function for Reissner-Nordstöm
and regular black holes. We also cover recent developments related to the
late-time attractors induced by Hawking radiation which exorcise the
exponential growth of the spacetime curvature encountered in the standard
mass-inflation scenario. In order to make the exposition self-contained, we
also briefly discuss basic properties of regular black holes including their
thermodynamics.

##  1 Introduction

To an external observer black holes are extremely simple objects. In the wake
of a gravitational collapse quadrupole moments and all the deformations
produced by the star either get swallowed inside the event horizon or are
carried away by gravitational radiation. At late times, the external field
settles into a Kerr-Newman geometry and is completely described by its mass,
charge, and angular momentum.

The stationary exterior field hides a rather complicated dynamics in the black
hole interior which drives spacetime towards its final classical fate,
determined by Penrose’s celebrated 1965 theorem Penrose:1964wq . The key
property of the black hole interior is that the Schwarzschild radial
coordinate $r$ becomes time-like within the event horizon. Thus a descent into
a black hole is a progression in time. The inner layers do not only enclose
but actually precede the core. This peculiar causal structure allows us to
theoretically explore the interior by boring in “layer-by-layer” starting from
shells situated at larger radii. The causal structure guarantees that what we
learn about the outer zone, where the classical description of the geometry is
still possible, cannot be affected by our ignorance about quantum gravity
effects potentially operating in the innermost regions close to the spacetime
singularity. From this point of view, it is possible to speak of an
“evolution” with increasing advanced time and subsequent relaxation to its
final state. The final “big-crunch” can then be imagined as an abrupt stop or
a rupture in the classical geometry as, accordingly to the strong cosmic
censorship, the inner singularity is space-like Penrose:1969pc .

In the case of no angular momentum the static configuration approached outside
the horizon is mirrored in the interior by an almost spherically symmetric
configuration where the internal disturbances are exponentially damped. As the
radial coordinate plays the role of time, it is conceivable that, before the
singularity, there exists a layer where the dynamics admits a semiclassical
description in terms of an effective Einstein equation originating from
quantum field theory in a curved spacetime Birrell:1982ix

$G_{\mu\nu}=8\pi G\,\langle T_{\mu\nu}\rangle\,.$ (1)

Here $G_{\mu\nu}$ is the (classical) Einstein tensor, $G$ is Newton’s
coupling, and $\langle T_{\mu\nu}\rangle$ is the (suitably regularized)
expectation value of the stress-energy tensor. Using Schwarzschild
coordinates, the sign of $\langle{T_{t}}^{t}\rangle$ dictates the fate of the
singularity. If it is negative, quantum polarization effects have a self-
regulatory effect. The Misner-Sharp quasi-local mass, formally introduced in
eq. (4), behaves like $M(r)\sim r^{3}$ near Planckian distances. This in turn
implies a de Sitter core inside the black hole Poisson:1988wc .

The idea of replacing the spacetime singularity by a patch of de Sitter space
actually comes with a long history. It was proposed in bardeen1968non ,
introduced in 1982ZhPmR..36..214M  in the context of the limiting-curvature
conjecture and further developed in dy92 ; frolov98 , also see ansoldi  for a
review. The formation and evaporation of these regular black holes has been
first discussed in hayward . From a fundamental perspective these types of
regular black holes have been motivated based on the gravitational asymptotic
safety program Bonanno:2000ep ; Bonanno:2006eu  (also see Cai:2010zh ;
Reuter:2010xb ; Koch:2013owa ; Koch:2014cqa ; Torres:2014gta ; Bonanno:2017zen
; Pawlowski:2018swz ; Adeifeoba:2018ydh ; Platania:2019kyx  for selected
follow-up works) and as Planck stars Rovelli:2014cta  (further explored in
Barrau:2014hda ; DeLorenzo:2014pta ; Christodoulou:2016vny  and reviewed in
Perez:2017cmj ) inspired by loop quantum gravity.

The de Sitter core shared by these models has important consequences for their
causal structure, since it implies the presence of an inner horizon, a so-
called Cauchy horizon. This feature may be crucial when considering the
regularity of the geometry in the presence of perturbations. Already in 1968
Penrose noted penrose68  that the ingoing sheet of the inner horizon,
corresponding to infinite advanced time, is a surface of infinite blueshift
for a wavelike disturbance propagating inwards. In particular, the radiative
tail of a generic collapse experiences an exponential blueshift close to the
Cauchy horizon. Most likely, this property signals the presence of an
instability.

This possibility was then explored in several perturbative studies. The first
investigation of the backreaction of the blueshifted influx onto the geometry
near the Cauchy horizon was performed by Hiscock 1981PhLA…83..110H , but it
was Poisson and Israel poisson89  and Ori ori91  who demonstrated that the
combined effect of influx and outflux produces an exponential growth of the
Coulomb component of the Weyl curvature as a function of the advanced time
coordinate $v$. This has been dubbed the “mass-inflation effect”. For an
astrophysical (non-zero angular momentum) black hole this effect has dramatic
consequences: at variance with the strong singularity at the center, the
curvature reaches planckian levels even if the radius of the inner horizon is
macroscopically large. Semiclassical quantum corrections on a dynamically
inflating geometry cannot really halt the build-up of this singularity. It
turns out though that the new singularity is weak in the Tipler sense. For
this reason, some authors have speculated that a $C^{1}$-extension of the
geometry is not ruled out Burko:1995uq . The arguments suggesting a possible
continuation of spacetime beyond the Cauchy horizon carry over to the case of
a regular black hole. In this case, the Cauchy horizon may be located at
planckian distances from the center and only a complete and consistent theory
of quantum gravity may be able to deliver a final answer on the possibility of
determining plausible extensions of the geometry though.

The relevant question is if it is possible to identify a dynamical mechanism
damping of the mass-inflation singularity which operates near the central
nucleus, despite our ignorance of the details concerning the structure of the
spacetime near the center. In fact, the late advanced time geometry can be
described in clearer physical terms because the initial data is well known in
this case. It is composed of two elements: the influx of gravitational waves
transmitted by the outer potential barrier inside the black hole and the
Hawking flux. The decay of the mass associated with the former is well-
described by an inverse power law of the type $1/v^{p-1}$ price72a , the so-
called Price’s tail ($p=12$ for a quadrupole moment). The latter contribution
has been conjectured in 1991PhLA..161..223B  and it was argued that this
should stop the growths of the mass function at early advanced times already
for mini black holes of mass $<100$ kg, approximately.

In the case of regular black holes the Hawking flux has a dramatic impact on
the final state of the geometry: it is given by an extremal configuration
which is reached in an infinite amount of advanced time. The problem of the
stability of the inner structure of a regular black hole then turns into a
delicate dynamical question with two competing time scales, the one associated
with the mass-inflation instability and the characteristic time-scale of the
evaporation. If the latter is much longer than the dynamical time-scale of the
instability, one can completely neglect the effect of the Hawking radiation
unless the black hole is of very low mass. On the other hand recent
calculations showed Bonanno:2020fgp  that for specific types of regular black
holes the exponential instability is turned into a much milder power-law
growth of the curvature near the Cauchy horizon. In this case one has to asses
the impact of the Hawking flux on the instability, using a dynamically
evolving background which is running towards its critical state in the
$v\rightarrow\infty$ limit. This analysis was recently performed in
Bonanno:2022jjp  and discovered two new classes of late-time attractors
governing the dynamics of mass-inflation at asymptotically late times.

The goal of this chapter is to review the mass-inflation effect for regular
black hole geometries in case of a static background as well as in the
presence of the Hawking flux. In order to make the discussion self-contained
we review the key properties of spherically symmetric spacetimes and regular
black holes in Sects. 2 and 3 respectively. Sect. 4 covers the “classical”
analysis of the mass-inflation effect for static Reissner-Nordström black
holes. Sect. 5 contrasts this situation to the mass-inflation effect
encountered for regular black holes. Our conclusion that _regular black holes
come with stable cores_ is detailed in Sect. 6. Throughout this chapter we
work in the probe-approximation, neglecting the back-reaction of the geometry
onto perturbations.

##  2 Spherically symmetric spacetimes: a primer

We start by summarizing the relevant properties of spherically symmetric
spacetimes. Throughout the discussion we adopt geometric units $G=c=1$. A
line-element exhibiting spherical symmetry can be cast into the form

$ds^{2}=g_{ab}dx^{a}dx^{b}+r^{2}d\Omega^{2}\,,\qquad a,b=0,1\,.$ (2)

Here, $x^{a}$ is any pair of coordinates that labels the set of two-spheres,
$d\Omega^{2}\equiv d\theta^{2}+\sin^{2}(\theta)d\varphi^{2}$ is the metric on
a unit sphere and $r(x^{a})$ is a function of $x^{a}$ defined by the geometric
condition that the area $A$ of the two-spheres is given by $A=4\pi r^{2}$. We
then define $f(x^{a})$ through the gradient

$f(x^{a})\equiv g^{ab}(\partial_{a}r)(\partial_{b}r)\,.$ (3)

The function $f(x^{a})$ is related to the Misner-Sharp mass $M(x^{a})$ through
the definition

$f(x^{a})=1-\frac{2M(x^{a})}{r}\,.$ (4)

A convenient choice for the coordinates $x^{a}$ is ingoing
Eddington–Finkelstein coordinates $x^{a}=(v,r)$ where (2) takes the form

$ds^{2}=-f(v,r)\,dv^{2}+2drdv+r^{2}d\Omega^{2}\,.$ (5)

We then use a dot to denote a derivative with respect to $v$,
$\dot{f}\equiv\frac{\partial f(v,r)}{\partial v}$, while derivatives with
respect to $r$ are indicated by primes, $f^{\prime}\equiv\frac{\partial
f(v,r)}{\partial r}$.

Starting from (5), it is instructive to express the most common curvature
scalars in terms of $M(v,r)$ and its derivatives. For the Ricci scalar $R$ one
readily finds

$R=\frac{2rM^{\prime\prime}+4M^{\prime}}{r^{2}}\,.$ (6)

In addition, we define the two curvature scalars appearing at second order of
the spacetime curvature. The Kretschmann scalar $K\equiv
R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$ and the square of the
Weyl-tensor $C^{2}\equiv
C_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta}$ evaluate to

$\begin{split}K=&\,\frac{16MM^{\prime\prime}}{r^{4}}+\frac{4(M^{\prime\prime})^{2}}{r^{2}}-\frac{64MM^{\prime}}{r^{5}}+\frac{32(M^{\prime})^{2}}{r^{4}}-\frac{16M^{\prime}M^{\prime\prime}}{r^{3}}+\frac{48M}{r^{6}}\,,\\\
C^{2}=&\frac{4\left(r^{2}M^{\prime\prime}-4rM^{\prime}+6M\right)^{2}}{3r^{6}}\,.\end{split}$
(7)

Since the geometries considered in this work are in general not Ricci flat,
$R_{\mu\nu}\not=0$, $K$ and $C^{2}$ are not identical and capture different
curvature properties. For later reference we also define the Coulomb-component
$\Psi_{2}\equiv-\frac{1}{2}{C^{\theta\varphi}}_{\theta\varphi}$ of the Weyl-
tensor which evaluates to

$\Psi_{2}=-\frac{M}{r^{3}}+\frac{2M^{\prime}}{3r^{2}}-\frac{M^{\prime\prime}}{6r}\,.$
(8)

For static geometries, where $f(v,r)$ is a function of $r$ only, the location
of the horizons is determined by the condition

$f(r)=0\,,\qquad r>0\,.$ (9)

For the generic situation discussed in this review, this equation has two
solutions. The event horizon (EH) is located at $r_{+}$. In addition, there is
a Cauchy horizon (CH) at $r_{-}<r_{+}$. The surface gravity $\kappa_{\pm}$ at
these points is defined as

$\kappa_{\pm}\equiv\pm\left.\frac{1}{2}\frac{\partial f(r)}{\partial
r}\right|_{r=r_{\pm}}\,.$ (10)

The choice of sign ensures that $\kappa_{\pm}>0$. In non-static spacetimes the
solutions of (9) may depend on $v$ and thus constitute apparent horizons.

We exemplify the general setting for the static Reissner-Nordström
(RN)-geometry. In this case, eq. (3) is given by

$f(r)=1-\frac{2m}{r}+\frac{Q^{2}}{r^{2}}\,,$ (11)

where $m$ is the asymptotic mass of the configuration and $Q$ denotes the
charge of the black hole. In this case the EH and CH are situated at

$r_{\pm}=m\pm(m^{2}-Q^{2})^{1/2}\,.$ (12)

The corresponding surface gravity is given by

$\kappa_{\pm}=\frac{m^{2}-Q^{2}}{r_{\pm}^{2}}\,.$ (13)

For $Q=m$ the position of the horizons coincide and one has the extremal RN
black hole. For this configuration, the surface gravity (13) vanishes.

##  3 Regular black holes

Upon completing our review of spherically symmetric geometries, we briefly
summarize the key properties of regular black holes. The idea of a regular de
Sitter core is introduced in Sect. 3.1 and various models implementing this
idea are reviewed in Sect. 3.2. We close with a short discourse on the
thermodynamical properties of regular black holes in Sect. 3.3.

###  3.1 de Sitter cores

The idea of a de Sitter core regularizing the interior of a Schwarzschild
black hole has been implemented in various settings. It is therefore
instructive to start by reviewing the general idea, also see Frolov:2016pav
for a general discussion. Let us consider a static, spherically symmetric
geometry characterized by the Misner-Sharp mass $M(r)$ introduced in eq. (4).
The general expressions for the Ricci scalar $R$ and the Kretschmann scalar
$K$ for this case have been given in eqs. (6) and (7). Demanding that the
curvature scalars remain finite as $r\rightarrow 0$ implies that $M(r)$ cannot
be constant. Regularity demands that

$\lim_{r\rightarrow 0}\,r^{-3}M(r)=\text{const}\,.$ (14)

Assuming a polynomial expansion around $r=0$, regularity then implies

$M(r)\simeq a_{3}\,r^{3}+a_{4}\,r^{4}+a_{5}\,r^{5}+{\cal O}(r^{5})\,,$ (15)

where $a_{i}$, $i=3,4,\cdots$ are real coefficients. Substituting this
expansion into eq. (4) gives

$f(r)\simeq 1-2a_{3}\,r^{2}-2a_{4}\,r^{3}-2a_{5}\,r^{4}+{\cal O}(r^{5})\,.$
(16)

For $a_{3}>0$ the local geometry at $r=0$ is the one of de Sitter space.111For
$a_{3}<0$ one encounters a regular anti-de Sitter core. Regular geometries of
this type have recently been discussed in the context of Gauss black holes
Boos:2021kqe . The loop black hole suggested in Modesto:2008im  also falls
into this classification having $a_{3}=0$. Combining the asymptotics (16) with
the condition of asymptotic flatness shows that a black hole with a de Sitter
core must come with an even number of horizons. This result is independent on
the adopted field equations. It follows from the analytical properties of
$M(r)$ near the origin. In static spacetimes with only two horizons, the
second, inner horizon is a Cauchy horizon. For non-static spacetimes, the
regularity condition implies that the apparent horizon cannot cross $r=0$.

###  3.2 Examples of regular black hole geometries

Regular black holes with a de Sitter core have been constructed by several
authors. Prominent examples are the Bardeen black hole bardeen1968non ; Ayon-
Beato:2000mjt , regular black holes constructed by Dymnikova dy92 ;
Dymnikova:2001fb , the Hayward black hole hayward , regular black holes
constructed within the gravitational asymptotic safety program Bonanno:2000ep
, and Planck stars motivated from loop quantum gravity Rovelli:2014cta ;
Saueressig:2015xua . For the sake of conciseness, we will limit the discussion
to the Hayward (H) geometry and the regular black holes found in the
gravitational asymptotic safety program (RG-improved black holes).

t]

Figure 1: Illustration of the horizon structure of the Hayward black hole
(17). For $m>m_{cr}$ one encounters an outer event horizon and an inner Cauchy
horizon. These horizons merge for $m=m_{cr}$. For $m<m_{cr}$ no horizons
appear. The graph has been obtained for a Hayward geometry with $l=1$.

_Hayward black holes._ Perhaps the simplest implementation of the above
requirements has been proposed by Hayward in hayward . In this case, the
function $f$ is given by

$f=1-\frac{2M(r)}{r},\quad\quad M(r)=\frac{mr^{3}}{r^{3}+2ml^{2}}\,.$ (17)

Here $l$ is a characteristic scale of the order of Planck length.
Interestingly, (17) was actually first derived by Poisson and Israel
Poisson:1988wc  assuming a simple relation between vacuum energy and curvature
via the effective Einstein field equations (1). In the “evolutionary" picture
of the interior described in the introduction one would then expect that a
consistent theory of quantum gravity should be able reproduce a similar
behavior at smaller radii.

Following the general discussion of Sect. 2, the explicit form of the horizon
condition (9) for the static Hayward geometry is

$2l^{2}m-2mr^{2}+r^{3}=0\,.$ (18)

For $m>m_{\rm cr}$ this equation has two real, positive roots $r_{+}>r_{-}$,
i.e., one encounters an event and one Cauchy horizon. For the critical mass
$m_{\rm cr}=\frac{3\sqrt{3}}{4}l$, the two horizons coincide and the black
hole is extremal. For $m<m_{\rm cr}$ one has a regular geometry without
horizons. This characteristic structure of $f(r)$ is illustrated in Fig. 1. It
is universal in the sense that it is essentially identical for all regular
black holes building on a de Sitter core.

It is also straightforward to evaluate the curvature scalars (7) for the
Misner-Sharp mass (17). The result is shown in Fig. 2 for masses $m$ given by
multiples of $m_{\rm cr}$. The Kretschmann scalar reaches its maximum at $r=0$
and decreases monotonically with increasing $r$. The square of the Weyl tensor
exhibits zeros at the origin and one specific point located between the
horizons $r_{-}$ and $r_{+}$. Thus, the regular _static_ black hole geometry
is compatible with the limiting curvature hypothesis markov1982ultimate ;
frolov1989through ; frolov1990black  for all values $m$.

_RG-improved black holes._ Interestingly, the gravitational asymptotic safety
program Percacci:2017fkn ; Reuter:2019byg  has provided non-trivial hints that
the theory supports regular black holes with a de Sitter core. Ref.
Bonanno:2000ep  applied the method of renormalization group (RG)-improvement
to a Schwarzschild black hole in order to obtain an effective geometry taking
quantum gravity corrections into account. This leads to a Misner sharp mass
exhibiting singularity resolution. In this case, the effective running of the
Newton constant at high energies produces an effective mass $M(r)$ which
vanishes as $r^{3}$ at small distances. The essential elements of the
construction, leading to the geometry (20), can be summarized as follows.

Figure 2: Illustration of the Kretschmann scalar $K$ (left) and the squared
Weyl tensor $C^{2}$ (right) evaluated for the Hayward geometry (17) with
$m=m_{\rm cr}$ (blue line), $m=2m_{\rm cr}$ (orange line), and $m=5m_{\rm cr}$
(green line). The curvature invariants are finite everywhere and the geometry
obeys the limiting curvature hypothesis. The graphs have been obtained for
$l=1$.

Investigating asymptotic safety based on solutions of the Wetterich equation
adapted to gravity Reuter:1996cp  leads to a scale-dependent Newton’s coupling
$G(k)$ depending on a coarse graining scale $k$. Using the arguably simplest
approximate solution of the Wetterich equation based on the Einstein-Hilbert
truncation Reuter:1996cp ; Lauscher:2001ya ; Reuter:2001ag , this scale-
dependence can be approximated by Bonanno:2000ep

$G(k)=\frac{G}{1+\omega\;G\;k^{2}}\,.$ (19)

Here $G$ is the laboratory value of Newton’s coupling measured at $k=0$ and
$\omega$ is a _positive_ constant. At large distances, $k\rightarrow 0$,
$G(k)$ approaches $G$. At short distances, $k\rightarrow\infty$, the scaling
$\lim_{k\rightarrow\infty}k^{2}G(k)=\omega^{-1}=g_{*}>0$ is dictated by the
Reuter fixed point providing the UV-completion of the theory.

Following the original derivation Bonanno:2000ep , the RG-improvement process
starts from the function $f(r)$ describing a classical Schwarzschild black
hole. Subsequently $G\mapsto G(k)$ is promoted to the scale-dependent coupling
(19). The RG-improved geometry is then obtained by identifying the coarse-
graining scale $k$ with the inverse radial proper distance $d(r)$ between the
origin and a point located at radius $r$, $k^{2}=\xi^{2}/d(r)^{2}$. This
procedure leads to the geometry Bonanno:2000ep

$f(r)=1-\frac{2Gmr^{2}}{r^{3}+{\widetilde{\omega}}G(r+\gamma mG)}\,,\qquad
M(r)=\frac{mr^{3}}{r^{3}+{\widetilde{\omega}}G(r+\gamma mG)}\,.$ (20)

Here $\widetilde{\omega}\equiv\omega\xi^{2}$ is a positive constant and
$\gamma\approx 9/2$ but the qualitative features of the model are independent
of its precise value. We now return to geometric units $G=1$. Moreover, we
will set $\gamma=0$. While this does not strictly correspond to a regular
black hole with a de Sitter core in the sense of eq. (14). Nevertheless, this
limit still captures all essential features of the horizons exhibited by the
regular model while significantly simplifying the discussion. The case
$\gamma=9/2$ can be found in Bonanno:2000ep .

Qualitatively, the features of the function $f(r)$ describing the RG-improved
black hole are identical to the ones shown in Fig. 1. The horizon condition
resulting from (20) is $(\gamma=0)$

$r\,\left(r^{2}-2mr+\widetilde{\omega}\right)=0\,.$ (21)

Analogous to the Hayward case, there exists a critical mass value

$m_{\rm cr}=\sqrt{\widetilde{\omega}}\,.$ (22)

For $m>m_{\rm cr}$, $f(r)$ has two simple zeros at

$r_{\pm}=m\pm\sqrt{m^{2}-m_{\rm cr}^{2}}\,.$ (23)

Hence, the spacetime has an outer event horizon at $r_{+}$ and an inner
(Cauchy) horizon at $r_{-}$. For $m=m_{\rm cr}$ the two horizons coincide and
there is one double zero at $r_{+}=r_{-}=\sqrt{\widetilde{\omega}}$ and the
black hole is extremal. If $m<m_{\rm cr}$, the spacetime is free from any
horizon.

###  3.3 Hawking effect and final state

So far, our exposition has been limited to static geometries. Following
Hawking’s seminal work Hawking:1975vcx , it is expected that black holes emit
thermal radiation in form of a Hawking flux. This radiation comes with a
perfect black body spectrum with temperature

$T_{\rm BH}=\frac{\kappa_{+}}{2\pi}\,,$ (24)

where $\kappa_{+}$ is the surface gravity at the event horizon (10). The
resulting energy loss leads to a decrease of the black hole’s mass. For a
Schwarzschild black hole with just an event horizon this entails that the
black hole evaporates completely within a finite proper time interval. This
can be traced back to the temperature being inversely proportional to the mass
$m$, so that the black hole turns hotter the lighter it gets.

The presence of a Cauchy horizon changes this picture drastically. While the
black hole is still expected to experience mass loss due to the Hawking flux,
the temperature of the radiation remains finite since the two horizons
approach each other as the black hole becomes lighter, c.f. Fig. 1. As a
consequence, the Hawking temperature decreases in the final stage of the
evaporation process and the geometry asymptotes to a cold remnant given by the
extremal black hole. Figure 3 contrasts these two situations. The black hole
evaporation of a Schwarzschild black hole is shown as orange lines indicating
that the process terminates in a finite time-span. The generic situation
encountered for regular black holes with a Cauchy horizon is shown by the blue
curves, illustrating that one obtains a remnant with mass $m_{\rm cr}$.

Figure 3: Illustration of the thermodynamical properties of a regular black
hole. Its temperature (left) remains finite and the mass (right) approaches
the critical mass $m_{cr}$ (dashed line) asymptotically. The graphs have been
obtained for a Hayward geometry with $l=1$. The Schwarzschild geometry ($l=0$)
has been added as the orange line for comparison.

We illustrate these properties using the technically simplest setting of the
RG-improved geometry (20) with $\gamma=0$. The analysis of other regular black
hole geometries featuring a Cauchy horizon follows along the same lines but
results in expressions which are significantly more complex. Thus we opt for
this example for pedagogical reasons. In order to stress the universal
features of the construction, we replace the model parameter
$\widetilde{\omega}$ by $m_{\rm cr}$ using (22).

Starting from (20) and evaluating (24) gives the black hole temperature as a
function of its mass

$T_{\rm BH}(m)={1\over 4\pi m}\;{\sqrt{1-\Omega^{2}}\over
1+\sqrt{1-\Omega^{2}}}\,,\qquad\Omega\equiv\frac{m_{cr}}{m}\,.$ (25)

The temperature vanishes for $m\searrow m_{\rm cr}$, i.e., $\Omega\nearrow 1$.
This feature underlies the interpretation of an extremal black hole as a
“cold" remnant.

Given the temperature (25), the luminosity of the black hole can be estimated
via the Stefan-Boltzmann law $L=\sigma T^{4}$, where $\sigma=\pi^{2}/60$ for a
single, massless degree of freedom. The emitted power is obtained by
multiplying $L$ with the area of the event horizon $A\equiv 4\pi r_{+}^{2}$,

$P(m)=\sigma\left(4\pi r_{+}^{2}\right)T^{4}\,.$ (26)

Here $T$ and $r_{+}$ are understood as functions of $m$. The mass-loss is then
determined by solving the mass-loss formula

$\dot{m}=-P(m)\,.$ (27)

For the concrete example of the RG-improved black hole

$P(m)=\frac{\sigma}{(4\pi)^{3}\,m^{2}}\,\frac{\left(1-\Omega^{2}\right)^{2}}{\big{(}1+\sqrt{1-\Omega^{2}}\big{)}^{2}}\,.$
(28)

The final part of the evaporation process is described by those terms in the
above expressions which are dominant for $m\searrow m_{\rm cr}$
($\Omega\nearrow 1$). Expanding (25) gives the asymptotic form of the
temperature

$T(m)\simeq\frac{\sqrt{m-m_{\rm cr}}}{2\sqrt{2}\pi\,m_{\rm cr}^{3/2}}\,,$ (29)

and the emitted power

$P(m)\simeq\frac{\sigma}{16\pi^{3}}\,\frac{(m-m_{\rm cr})^{2}}{m_{\rm
cr}^{4}}\,.$ (30)

Integrating (27) in the asymptotic regime then yields

$m(v)\simeq m_{\rm cr}+\frac{m_{0}-m_{\rm cr}}{1+\alpha(m_{0}-m_{\rm
cr})(v-v_{0})}$ (31)

Here $\alpha\equiv\sigma/(16\pi^{3}\widetilde{\omega}^{2})$, and $v_{0}$ is a
time, already in the late-time regime, where $m(v_{0})=m_{0}$ is imposed. The
key result is that for $v\rightarrow\infty$

$m(v)-m_{\rm cr}\propto 1/v\,.$ (32)

This result is universal, i.e. it does not depend on the details of the
geometry of the regular black hole Bonanno:2022jjp . Eq. (75) shows that it
carries over to the Reissner-Nordström geometry and the Hayward regular black
hole as well. The profound consequence of (32) arises from the comparison with
Price’s law

$m_{-}(v)\simeq m_{0}-\frac{\beta}{(v/v_{0})^{p-1}}\,,\qquad p\geq 12\,,$ (33)

where $m_{0}$ is the asymptotic mass of the black hole, $\beta>0$ is a
quantity with the dimension of a mass, and $v_{0}$ is the initial time which
we set to one in the sequel. This shows that the Hawking flux dominates over
the gravitational waves contribution which vanishes as $1/v^{p}$. Hence the
Hawking effect may play an important role for the stability of a regular black
hole at asymptotically late times.

In order to judge the phenomenological viability of regular black holes,
understanding whether these geometries are robust once perturbations are
included is highly relevant. This applies in particular to perturbations of
the asymptotically extreme geometry with respect to the combined influx of
Hawking radiation and outflux of the star. At this point we have all the
prerequisites to review the current understanding of this situation in the
next sections.

##  4 The elementary mechanics of mass-inflation

We start by reviewing the basic mechanisms underlying the mass-inflation
effect. Our discussion focuses on the historical analysis carried out in the
context of the spherically symmetric Reissner-Nordström geometry. The key
result of the analysis is that the mass function $m_{+}(v)$ in the vicinity of
the Cauchy horizon grows exponentially once perturbations which naturally
appear in a gravitational collapse of a star into a black hole are included,

$m_{+}(v)\sim v^{-(p-1)}e^{\kappa_{-}v}\,,\qquad\text{as}\quad
v\rightarrow\infty\,,$ (34)

with $p$ being the exponent appearing in Price’s law (33). We start by giving
a qualitative discussion of the mass-inflation mechanism in Sect. 4.1. Sect.
4.2 then derives (34) based on the two-dimensional wave equation for the
generalized mass function in the presence of a cross-flow of outgoing and
ingoing streams of radiation. We close with a brief discussion on the strength
of the resulting signularity in Sect. 4.3.

###  4.1 Colliding mass-shells and DTR-relations

Figure 4: Two spherically symmetric, transparent, concentric shells colliding
at the speed of light: (a) before collision, (b) after the collision. The blue
circle represent ingoing radiation close to the Cauchy horizon, while the red
shell represents the outgoing radiation.

Qualitatively, the divergence of the local mass in the interior of a realistic
black hole can be understood with the following example (see Hu:1995jr ). Let
us consider a time-like moving shell of radius $R(\tau)$ where $\tau$ is the
proper time. The shell divides spacetime into a region $\mathcal{M}_{+}$
inside and $\mathcal{M}_{-}$ outside the shell and we assume that the mass
functions $m_{+}$ and $m_{-}$ in each sector are constant. The proper mass of
the shell, $m_{\rm shell}$, satisfies

$dm_{\rm shell}+P\,d(4\pi R^{2})=0$ (35)

which implies mass conservation if $P=0$. The equation of motion of the shell
is ruled by 1970NCimB..67..136C

$m_{-}-m_{+}=m_{\rm
shell}\left(1-\frac{2m_{+}}{R}+\left(\frac{dR}{d\tau}\right)^{2}\right)^{1/2}-\frac{m_{\rm
shell}^{2}}{2R}$ (36)

It expresses the total conserved gravitating mass $m_{-}-m_{+}$ of the shell
as a sum of four terms (expanding the square root to first order): the rest-
mass $m_{\rm shell}$, the kinetic energy $\frac{1}{2}m_{\rm
shell}\dot{R}^{2}$, the mutual potential energy $-m_{\rm shell}m_{+}/R$, and a
self-potential energy $-\frac{1}{2}m_{\rm shell}^{2}/R$. Note that the
“potential energy” contributes to the total mass of the outer body, it is a
“binding energy” Blau:1989zs . If the outer body can be released from its
gravitational binding, its gravitating mass can increase.

This can be easily illustrated in the light-like limit. In this case, we may
gain intuition from the DTR-relations, first discovered by Dray, ’t Hooft, and
Redmount Dray:1985yt ; 1985PThPh..73.1401R , and subsequently generalized in
bar91 . In the latter work the (generalized) DTR-relations are derived as
geometrical consistency conditions on the function $f$, beyond spherical
symmetry, for an energy-momentum tensor which is vacuum except for a
distributional delta function source representing the shells, also see
Blau:1989zs  for an instructive discussion.

The idea behind the DTR-relations is to consider the collision of two
infinitely thin, light-like, pressureless shells describing an ingoing and
outgoing perturbation, see Fig. 4.222For more details on the infinitely thin-
shell formalism, see Israel:1966rt . The shells provide a highly idealized
model of matter perturbations in the black hole interior which are naturally
expected from the collapse of a star into a black hole. They capture the
essential features underlying the mass-inflation effect, the counter-streaming
of matter located between the inner and outer horizon of the black hole in the
optical geometric limit Hamilton:2008zz .

As depicted in Fig. 5, the ingoing and outgoing shells separate spacetime in
four regions
$\mathcal{M}_{A},\mathcal{M}_{B},\mathcal{M}_{C},\mathcal{M}_{D}$. The
generalized DTR-relations then encode the fact that the spacetime metric in
each region must agree at the collision point $r_{0}$. We then assume that in
each region the metric takes the form (5) with $f_{i}(v,r)=f_{i}(r)$,
$i=A,B,C,D$, being static. Following the pedagogical derivation given in
Brown:2011tv , this entails that

$f_{A}\,f_{B}=f_{C}\,f_{D}\,,\quad\text{at}\quad r_{0}\,.$ (37)

For the sake of simplicity, let us assume that the metric in each sector has
the form of the Schwarzschild metric. Assuming that the region
$\mathcal{M}_{C}$ corresponds to flat space, the functions $f_{i}$ in each
region take the form

$f_{C}=1\,,\qquad f_{A}=1-\frac{2m_{1}}{r_{0}}\,,\qquad
f_{B}=1-\frac{2m_{4}}{r_{0}}\,,$ (38)

together with

$f_{D}=1-\frac{2(m_{3}+m_{4})}{r_{0}}=1-\frac{2(m_{1}+m_{2})}{r_{0}}\,.$ (39)

The relation (39) implies the conservation law

$m_{1}+m_{2}=m_{3}+m_{4}\,.$ (40)

Moreover, evaluating (37) gives the additional conditions

$m_{1}=m_{3}(1-2m_{4}/r_{0})^{-1},\quad m_{2}=m_{4}(1-2m_{1}/r_{0})\,.$ (41)

We now consider a collision just outside the horizon of the interior field,
$r_{0}=2m_{4}+\delta$ with $\delta\rightarrow 0^{+}$. The first identity in
(41) then entails that, close to the inner horizon, the mass increase
$m_{1}-m_{3}$ diverges as

$m_{1}-m_{3}=\frac{2m_{3}m_{4}}{\delta}\,.$ (42)

This divergence implies that the new Schwarzschild mass of the outgoing shell
would then become negative because of its potential energy.

This example, albeit very simple, illustrates the basic physical mechanism
underling the mass-inflation phenomenon: the imploding shell can mimic the
fallout from the radiative tail of the collapse while the outgoing shell
models the outflow from the collapsing star as depicted in Fig. 4. In a
spacetime diagram the Cauchy horizon would develop beyond the history of
region A, extending region B and the collision point $r_{0}$ is near the
actual Cauchy horizon. The fact that the DTR-relations are not limited to
spherical symmetry then suggests that the same phenomenon also appears in less
symmetric situations including a Kerr black hole 1991PhLA..161..223B .

Figure 5: Fig. 4 recast as a spacetime diagram: $r_{0}$ is the colliding
radius and sectors $B$ and $C$ represent the future and past evolution of the
spacetime, respectively.

This is the best that can be concluded on the basis of DTR-relations. In
particular, promoting $m_{1}-m_{3}$ to a function of $v$ is beyond the
framework and should not be used to draw conclusions on the strength of the
singularity, albeit this point is sometimes not appreciated in the literature
Carballo-Rubio:2018pmi ; 2021PDU….3300853B . Determining the strength of the
divergence requires a more detailed analysis based on dynamical models. We
cover this in the next subsection.

###  4.2 Dynamical models of mass inflation

The simple analogy discussed in the previous section can be useful to
understand the interior of realistic black holes. In this case, it is useful
to make some simplifying assumptions. A realistic, rotating (non-spherical)
black hole may be schematized as a spherically symmetric, charged black hole
because their horizon structures are similar. Moreover, the tail of
gravitational quadrupolar waves may be idealized as spherical waves. Since the
perturbations get blue-shifted near the Cauchy horizon, one can use an
“optical” approximation and describe the matter infalling onto the Cauchy
horizon as a stream of light-like particles.

In 1981, Hiscock 1981PhLA…83..110H  treated the dynamics of this situation by
considering the charged Vaidya metric,

$ds^{2}=2drdv-\left(1-\frac{2m(v)}{r}+\frac{Q^{2}}{r^{2}}\right)dv^{2}+r^{2}d\Omega^{2}\,,$
(43)

promoting the mass function of the Reissner-Nordström metric to a
$v$-dependent function. Then $m(v)$ is the externally measured mass of the
black hole which varies with advanced time $v$ because of the inflow. The
associated stress-energy tensor (the electrostatic component is irrelevant in
this discussion) corresponds to radially infalling light-like dust:

$T_{\alpha\beta}=\frac{\dot{m}(v)}{4\pi r^{2}}l_{\alpha}l_{\beta},\quad
l_{\alpha}=-\partial_{\alpha}v\,,\quad l_{\alpha}l^{\alpha}=0\,.$ (44)

The ansatz

$m(v)=m_{0}-\frac{\beta}{(v/v_{0})^{p-1}}$ (45)

reproduces the power-law decay of the radiative tail. (For the generic case of
the “quadrupole” waves $p=12$.) It is easy to show from (44) and (45) that an
observer approaching the Cauchy horizon ($v\rightarrow\infty$) reaches the
horizon in a finite proper time and measures an energy flux diverging like
$e^{2\kappa_{-}v}$, with $\kappa_{-}$ being the surface gravity (13) for the
asymptotic, stationary Reissner-Nordström black hole of mass $m_{0}$ and
charge $Q$.

In Poisson:1990eh , Poisson and Israel extended this model by including a
crossflow of radially ingoing and outgoing radiation. This analysis shows that
the geometry near the Cauchy horizon changes dramatically. Approaching the
Cauchy horizon, the Coulomb component of the Weyl curvature

$\Psi_{2}=-\frac{m(v)}{2r^{3}}+\frac{Q^{2}}{r^{4}}\,,$ (46)

and hence the mass parameter $m(v)$ diverge like

$m(v)\simeq v^{-(p-1)}e^{\kappa_{-}v},\quad v\rightarrow\infty\,.$ (47)

This behavior has to be contrasted with the situation outside the event
horizon, where $m$ approaches its ADM value at infinity.

It is interesting to see how this effect originates from the point of view of
the field equations. A spherically geometry can be described by the metric
(2). The Einstein equations can be reformulated as a two-dimensional,
covariant wave equation for $m$,

$\Box\,m=-16\pi^{2}r^{3}T^{ab}T_{ab}\,.$ (48)

Going to Kruskal coordinates, the energy momentum tensor for a null-
crossflowing radiation can be written as

$T_{ab}=\frac{L_{\rm in}(V)}{4\pi
r^{2}}\partial_{a}V\partial_{b}V+\frac{L_{\rm out}(U)}{4\pi
r^{2}}\partial_{a}U\partial_{b}U\,.$ (49)

Due to $\partial_{a}V$ and $\partial_{a}U$ being null vectors, the square of
the energy momentum tensor vanishes if only one component $T^{\rm out}_{ab}$
or $T^{\rm in}_{ab}$ is considered. In this case the source on the right-hand
side of eq. (48) vanishes and there is no mass-inflation. It is only when
_both components_ $T^{\rm out}_{ab}$ or $T^{\rm in}_{ab}$ are present that the
wave equation contains a non-trivial source term triggering the growth of $m$.

At this stage, one has to adopt assumptions for $L_{\rm in}$ and $L_{\rm
out}$. $L_{\rm in}$ uses Price’s law (45), which in Kruskal coordinates $V$
reads

$L_{in}(V)=\frac{\beta}{(-\kappa_{-}V)^{2}}(-\ln(-\kappa_{-}V))^{-p}.$ (50)

$L_{\rm out}(U)$ receives two contributions, one from the collapsing star and
a second one from gravitational waves reflected by the inner potential
barrier. At late advanced time the first one is negligible. The second one is
basically following Price’s law, but written in terms of the coordinate $U$.
Thus, we have control over $L_{\rm out}(U)$ at asymptotically late times only
and every analysis going beyond this regime has to track the full dynamics
numerically.

Near the Cauchy horizon,333This makes the crucial assumption that the Cauchy
horizon actually exists. While there is good evidence for forming such a
horizon for the black holes encountered in general relativity, the question
whether such a horizon actually forms in the collapse to a regular black hole
is a largely open question, see Ziprick:2010vb ; Biasi:2022ktq  for some
recent work in this direction. For a first self-consistent dynamical
calculation of mass-inflation without assuming the existence of a Cauchy
horizon we refer to Bonanno:1994qh . assuming that $r\approx r_{0}$, equation
(48) then reads

$\partial_{U}\partial_{V}m=-\frac{2}{r_{0}}L_{\rm in}(V)L_{\rm out}(U)\,.$
(51)

In the limit $V\rightarrow 0_{-}$ this relation is readily integrated,
producing a divergence of the type

$m\simeq\frac{1}{-\kappa_{-}V(\ln(-\kappa_{-}V))^{(p-1)}}\,.$ (52)

In terms of the original advanced time coordinate $v$, (52) entails the
exponential growths of the mass function underlying the mass-infaltion effect
(34).

###  4.3 Mass-inflation instability and singularities of spacetime

At this stage it is interesting to discuss the implications of the growths of
the mass function (34) for the singularity structure of spacetime. We recall
that a singularity for which all algebraic scalars of curvature are finite and
yet some component of the curvature diverge in some frame is known as a
“whimper” singularity. In most of the cases these singularities are unstable
and tend to evolve in a strong, “scalar” singularity. As demonstrated in the
previous section, it turns out that if the Hiscock model is perturbed with an
additional outflow of radiation, which in a realistic collapse would always be
present either for the presence of the collapsing star or for the scattering
gravitational waves from the inner potential barrier, the Weyl curvature
diverges without limit at the Cauchy horizon.

The strength of the resulting singularity can then be understood as follows.
We focus on the future sector of the shell, denoting the coordinates and mass
function in this sector with the subscript “$+$”. Using the advanced
coordinate, the asymptotic form of the metric near the Cauchy horizon reads

$ds^{2}\simeq
2\frac{dv_{+}}{r}\left(rdr+m_{+}(v_{+})dv_{+}\right)+r^{2}d\Omega^{2}\,.$ (53)

We then define a new coordinate $u$ through the relation
$du=(rdr+m_{+}(v_{+})dv_{+})$. This coordinate is regular at the Cauchy
horizon. The line-element (53) then becomes Bonanno:1994qh

$ds^{2}\simeq 2\frac{dv_{+}du}{r}+r^{2}d\Omega^{2}\,.$ (54)

Again, this expression is manifestly regular at the Cauchy horizon. Since it
is possible to find a coordinate system where the metric is regular, the
singularity building up at the Cauchy horizon is rather weak. This fact has
profound consequences: as already realized by Ori ori91  and further
investigated by Burko Burko:1999zv , the mass-inflation singularity does not
satisfy the necessary conditions to be strong in the Tipler sense
Tipler:1977zza .444According to Tipler, a null singularity is called “strong”
if there exists at least one component of the Riemann tensor (in a parallelly
propagated frame) which does not converge when integrated with respect to the
affine parameter $\tau$ twice. The physical meaning of this requirement is
that the tidal distortion is not finite as an observer crosses the
singularity. A measure of the tidal distortion experienced by an observer is
obtained by integrating the square of the Weyl curvature twice. In the case of
the standard mass-inflation scenario one finds

$(\Psi_{2})^{2}\simeq
C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}\simeq\frac{1}{(\kappa_{-}V)^{2}(\log(-\kappa_{-}V))^{2(p-1)}}\,.$
(55)

Here $\Psi_{2}$ is the Coulomb-component of the Weyl curvature (46) and
$V\propto\tau$ is proportional to the proper time of an observer impacting on
the horizon. The tidal distortion is obtained by twice integrating (55) and is
therefore finite. It has further been argued by Ori that this behavior could
be sufficient to determine a $C^{1}$-extension of the spacetime beyond the
Cauchy horizon ori91 . However, according to Królak, eq. (55) still signals a
strong singularity, as the expansion of the congruence is divergent
1987JMP….28.2685K : if the components of the Riemann tensor are integrated
only once, the integral does not converge on the singularity.

##  5 Mass-inflation for regular black holes

We now embark on the central theme of this chapter, contrasting the mass-
inflation scenario for Reissner-Nordström and regular black hole geometries.
In the latter case we use the Hayward geometry as an explicit representative.
The results are more general though, since the Reissner-Nordström and Hayward
geometry constitute representatives for the two classes of universal late-time
behavior encountered in the literature Bonanno:2020fgp ; Bonanno:2022jjp . The
discussion will be based on the Ori-model introduced in Sect. 5.1. The case of
a static background is covered in Sect. 5.2 while the modifications due to the
mass-loss generated by the Hawking effect are highlighted in Sect. 5.3. In
order to contrast the structural differences of the mass-inflation effect for
the Reissner-Nordström case and regular black holes, we discuss the two
geometries in parallel.

###  5.1 The Ori-model – general setup and dynamics

The Ori-model constitutes a simplification of the original Poisson-Israel
model geared towards making the mass-inflation effect accessible by analytic
methods. In this case the outgoing energy flux is modelled by a spherically
symmetric, pressureless null shell $\Sigma$ placed between the (apparent)
inner and outer horizon of the black hole geometry. In the language of eq.
(49) it corresponds to taking $L_{\rm out}(U)$ to be a delta-function. The
shell then acts as a catalyst triggering the mass-inflation instability. In
this section, we derive the equations capturing the dynamics of the system for
a generic, spherically symmetric black hole spacetime exhibiting an event and
a Cauchy horizon.

The shell $\Sigma$ modeling the ingoing perturbation, divides spacetime into a
region $\mathcal{M}_{+}$ inside and $\mathcal{M}_{-}$ outside the shell.
Denoting the coordinates on $\mathcal{M}_{\pm}$ with subscripts $\pm$, the
metric in each sector can be written as

$ds^{2}=-f_{\pm}(r,v_{\pm})dv_{\pm}+2drdv_{\pm}+r^{2}d\Omega^{2}\,.$ (56)

Insisting that both regions lead to the same induced metric on $\Sigma$ yields
that the radial coordinate $r$ can be taken the same in both regions. Eq. (56)
already anticipated this result. In general, the relation between $v_{+}$ and
$v_{-}$ is non-trivial though. Their dependence is fixed by noting that the
position of $\Sigma$ in the two coordinate systems is

$f_{+}\,dv_{+}=f_{-}\,dv_{-}\,.$ (57)

In practice, we use this relation to express all quantities in terms of the
coordinate in the outer sector of the shell, setting $v\equiv v_{-}$. Eqs.
(56) and (57) are independent of the dynamics and completely fixed by the
geometrical setup of the model.

Our next task is to find the relation between the Misner-Sharp mass $M_{\pm}$
in the two sectors. Evaluating Einstein’s equations in each sector implies

$\frac{\partial M}{\partial r}=-4\pi r^{2}T_{v}{}^{v}\,,\qquad\frac{\partial
M}{\partial v}=4\pi r^{2}T_{v}{}^{r}\,,$ (58)

combined with $T_{rr}=0$. We then introduce the null generators of $\Sigma$

$s_{\pm}^{\mu}\equiv\frac{dx^{\mu}_{\pm}}{dr}=\left(2/f_{\pm},1,0,0\right)\,,$
(59)

where it is convenient to use $r$ as a parameter. Continuity of the flux
across $\Sigma$ requires

$\left[T_{\mu\nu}\,s^{\mu}s^{\nu}\right]=0\,$ (60)

where the square brackets indicate the discontinuity of a scalar quantity
across the position of the shell. In terms of the lapse- and the mass-
functions, eq. (60) implies

$\left.\frac{1}{f_{+}^{2}}\frac{\partial M_{+}}{\partial
v_{+}}\right|_{\Sigma}=\left.\frac{1}{f_{-}^{2}}\frac{\partial M_{-}}{\partial
v_{-}}\right|_{\Sigma}\,.$ (61)

We then recast this equation in terms of the coordinate $v$. Using, eq. (57),
it is convenient to write

$\left.\frac{1}{f_{+}}\frac{\partial M_{+}}{\partial
v}\right|_{\Sigma}=F(v)\,,$ (62)

where

$F(v)\equiv\left.\frac{1}{f_{-}}\frac{\partial M_{-}}{\partial
v}\right|_{\Sigma}\,.$ (63)

The introduction of $F(v)$ will facilitate the analysis of the late-time
dynamics of the model later on.

Finally, we need an equation determining the dynamics of the shell. Exploiting
that $\Sigma$ moves light-like, eq. (56) gives the relation
$f_{-}\,dv_{-}=2dr$. Thus the position of the shell $R(v)$ follows from

$\frac{dR}{dv}=\left.\frac{1}{2}f_{-}\right|_{\Sigma}\,.$ (64)

Eqs. (62) and (64) then form a coupled dynamical system determining the
position of the shell and the Misner-Sharp mass in its interior in terms of
the mass-function $m_{-}$ in the outer sector of the shell. Since the dynamics
of $R(v)$ is independent of $M_{+}$, one can first study the motion of the
shell based on (64) before substituting the solution into (62).

The Ori-model applied to a static black hole background fixes $m_{-}$ by
imposing the Price’s tail behavior (33). The Price tail governs the decay of a
perturbation at asymptotically late times. We stress that the relation (33)
applies only asymptotically. At intermediate times one expects that the time-
dependence of $m_{-}(v)$ may be significantly more complicated than indicated
by the asymptotic relation.

###  5.2 The Ori-model on static backgrounds

The Misner-Sharp mass for the Reissner-Nordström (RN) and Hayward (H) black
hole is

$\begin{split}{\rm RN:}\qquad&\,M(r)=m-\frac{Q^{2}}{2r}\,,\\\ {\rm
H:}\qquad&\,M(r)=\frac{mr^{3}}{r^{3}+2ml^{2}}\,.\end{split}$ (65)

Here $Q$ is the charge of the black hole and $l>0$ is a parameter with the
dimension of mass which ensures the regularity of the Hayward geometry. The
static background analysis includes the effect of the perturbation by
identifying the mass function $m_{-}(v)$ in the outer sector of the shell with
the Price-tail behavior given in eq. (33).

Figure 6: Illustration of the shell dynamics (blue curves) for the background
spacetime given by a static Reissner-Nordström (left) and Hayward black hole
(right). The gray lines depict the positions of the apparent event (top) and
Cauchy horizons (bottom). The shells reach $r_{-}$ at asymptotically late
times. The model parameters $Q$ and $l$ have been chosen such that the
critical mass of the remnant is $m_{cr}=1$.

Specifying the general equations (62) and (64) to our exemplary geometries,
one finds that the dynamics of the shell is given by

$\begin{split}{\rm
RN}:\qquad\dot{R}(v)=&\,\frac{R^{2}-2m\,R+Q^{2}}{2R^{2}}\,,\\\ {\rm
H}:\qquad\dot{R}(v)=&\,\frac{R^{3}-2m\,R^{2}+2l^{2}\,m}{2\left(R^{3}+2l^{2}m\right)}\,.\end{split}$
(66)

The mass function $m_{+}(v)$ in the interior is determined from

$\dot{m}_{+}(v)=p(m_{+})\,F(v)\,.$ (67)

The explicit forms of the functions $F(v)$ are

$\begin{split}{\rm RN}:\qquad
F(v)=&\,\frac{\dot{m}_{-}}{R^{2}-2R\,m_{-}+Q^{2}}\,,\\\ {\rm H}:\qquad
F(v)=&\,\frac{\dot{m}_{-}}{\left(R^{3}+2l^{2}m_{-}\right)\left(R^{3}-2(R^{2}-l^{2})m_{-}\right)}\,,\end{split}$
(68)

while the dependence of the right-hand side on $m_{+}$ is captured by the
polynomials

$\begin{split}{\rm RN}:\qquad p(m_{+})=&\,R^{2}-2m_{+}\,R+Q^{2}\,,\\\ {\rm
H}:\qquad
p(m_{+})=&\,\left(R^{3}+2l^{2}m_{+}\right)\left(R^{3}-2(R^{2}-l^{2})m_{+}\right)\,.\end{split}$
(69)

The polynomials (69) encode the key difference of the two geometries: in the
Reissner-Nordström case $p(m_{+})$ is linear in the mass while for Hayward it
is quadratic. As it turns out, this makes a decisive difference in the
stability of the two models.

We proceed by analyzing the late-time dynamics of the two systems. At the
analytic level, this is conveniently done by employing the Frobenius method.
In the specific case at hand, the $v$-dependent functions are expanded in a
generalized power series in $1/v$ which take the general form

$f(v)=\frac{1}{v^{s}}\sum_{k=0}^{\infty}\,\frac{a_{k}}{v^{k/2}}\,.$ (70)

The parameter $s$ and the coefficients $a_{k}$ are obtained from substituting
this ansatz in the corresponding differential equation, performing an
expansion for large $v$, and extracting a hierarchy of equations given by the
coefficients appearing at each order in the large-$v$-expansion. This
hierarchy is then solved recursively for $s$ (lowest order equation) and the
coefficients $a_{k}$. Note that eq. (70) already anticipates that consistent
solutions require the inclusion of non-integer powers of $1/v$.

Based on this strategy, one finds the following late-time behavior. Starting
with the dynamics of the shell, one first establishes that $r_{-}$ is a fixed
point of (66) since $f_{-}|_{r=r_{-}}=0$ by definition of the Cauchy horizon.
The first correction term obtained from the Frobenius analysis shows that
$r_{-}$ is indeed a late-time attractor,

$R(v)\simeq r_{-}+cv^{-(p-1)}\,,$ (71)

The constant $c$ is a positive coefficient depending on $m_{0}$ and $\beta$
and again $\simeq$ denotes that the expression holds asymptotically for large
$v$. Integrating (66) numerically confirms this property. Some sample
solutions arising from imposing initial conditions at different values $v$ are
shown in Fig. 6. This confirms that the shell impacts on $r_{-}$ rather
quickly from $R(v)>r_{-}$ and then essentially keeps its position close to the
Cauchy horizon.

Figure 7: Top: Illustration of the mass function $m_{+}(v)$ inside the shell
for a static Reissner-Nordström (left) and Hayward black hole (right). For the
blue lines the numerical integration terminates at finite value $v$ while the
green lines reach the late-time attractor of the static Hayward geometry where
$m_{+}^{\infty}\equiv\lim_{v\rightarrow\infty}m_{+}(v)$ remains finite (black
dashed line). Botton: Illustration of the Kretschmann scalar evaluated at the
position of the shell. For the green solutions, the curvature scalar grows
polynomially in $v$. The dashed line added in the Hayward case gives the value
of $|K|$ at the point where the blue solutions terminate, indicating that the
curvature scalar remains finite. The model parameters $Q$ and $l$ have been
chosen such that the critical mass of the remnant is $m_{cr}=1$.

Based on the asymptotic solutions for $m_{-}(v)$ and $R(v)$ one then readily
deduces the asymptotic behavior of $F(v)$,

$F(v)\simeq-\frac{1}{2}\,r_{-}\,\kappa_{-}\,,$ (72)

with $\kappa_{-}>0$ the surface gravity at the Cauchy horizon. Remarkably, the
negative constant appearing in this relation is universal in the sense that it
is the same for both geometries. Eq. (72) then allows to conclude the
asymptotic behavior of $m_{+}(v)$. It is at this point, where the different
structures of $p(m_{+})$ enter, yielding

$\begin{split}{\rm RN:}\qquad&m_{+}(v)\simeq
c_{1}\,e^{\kappa_{-}v}\,v^{-(p-1)}\,,\\\ {\rm
H:}\qquad&m_{+}(v)\simeq-\frac{r_{-}^{3}}{2l^{2}}\,.\end{split}$ (73)

In the case of $p(m_{+})$ being linear, $m_{+}(v)$ grows exponentially in $v$,
justifying the terminology “mass-inflation”. For quadratic polynomials
$p(m_{+})$ admits a second class of solutions where $m_{+}(v)$ takes a finite,
negative value asymptotically. This value is given by the negative root of
$p(m_{+})$ and thus constitutes a fixed point of (67). The role of this
attractor is illustrated at the level of numerical solutions in the top row of
Fig. 7. Imposing initial conditions at $v=1$ (which may be outside of the
validity of the Ori-model) one finds that solutions for the Reissner-Nordström
geometry terminate at finite value $v$ (blue curves). In addition to this
behavior, the Hayward geometry admits a second class of solutions (green
curves): these extend to asymptotically late times and approach the attractor
(73).

In order to understand the physics consequences entailed by the late-time
behavior of the mass-function, it is instructive to study the growth of the
Kretschmann scalar (7) evaluated at the position of the shell. For the
asymptotic solutions (73) this yields

$\begin{split}{\rm RN:}\qquad&K|_{\Sigma}\propto
e^{2\kappa_{-}v}\,v^{-2(p-1)}\,,\\\ {\rm H:}\qquad&K|_{\Sigma}\propto
v^{6(p-1)}\,.\end{split}$ (74)

Thus the structure of $p(m_{+})$ _can lead to a significant weakening of the
mass-inflation effect for regular black holes_. The origin of the growths (74)
in the two cases is quite different though: while the exponential growths in
the Reissner-Nordström case is directly related to the growths of the mass-
function, the polynomial growths in the Hayward case is tracked back to the
fact that the quantity $(R^{3}+2l^{2}m_{+})$, which vanishes at the late-time
attractor, also appears in the denominator of the Kretschmann scalar. The
scaling laws (74) are illustrated in the bottom row of Fig. 7. For the blue
solutions $K|_{\Sigma}$ diverges at finite values $v$ without reaching the
late-time attractor. This is different for the green solutions where the
numerical integration confirms the polynomial growths of $K|_{\Sigma}$ in the
Hayward case.

The weakening of the curvature singularity (74) may have profound consequences
for the geodesic completeness of spacetime. Solving the geodesic equation for
a massive, radially free-falling observer in the static, non-critical black
hole spacetime shows that $v=\infty$ can be reached in a finite amount of the
observer’s proper time. Technically, the new late-time attractor turns the
mass-inflation singularity into a weak singularity with respect to both the
Tipler and the Królak definition. This may open the possibility to extend
geodesics beyond the singularity. While this is certainly an exciting
possibility, this point is currently still awaiting its final clarification.

###  5.3 The Ori-model including Hawking radiation

When discussing the thermodynamics of regular black holes in Sect. 3.2, we
argued that the final state of the evaporation process is a cold remnant with
finite mass $m_{cr}$. The mass-inflation effect associated with the Cauchy
horizon then raises the question whether this picture is robust against
perturbations. The analysis for static, regular black holes showed that there
are mechanisms which tame the growth of the curvature singularity, delaying
the growths of the Misner-Sharp mass to asymptotically late times. This
suggests that the Hawking effect can influence the dynamics. The analysis
based on the Ori-model Bonanno:2022jjp , summarized in this section, indeed
confirms this expectation.

Upon including the Hawking effect, the mass function of the black hole is no
longer constant but turns into a $v$-dependent function. The time-dependence
can be determined from the mass-loss formula (27). Following the strategy of
the previous section, focusing on the late-time behavior of the solutions, it
is straightforward to determine the leading terms from the Frobenius method

$\begin{split}{\rm RN:}\qquad&\,m_{\rm Hawking}(v)\simeq
m_{cr}\left(1+2\,\frac{15\pi\,m_{cr}^{3}}{v}+48\left(\frac{15\pi
m_{cr}^{3}}{v}\right)^{3/2}+\cdots\right)\,,\\\ {\rm H:}\qquad&\,m_{\rm
Hawking}(v)\simeq
m_{cr}\left(1+6\,\frac{80\pi\,m_{cr}^{3}}{v}+80\,\left(\frac{80\pi\,m_{cr}^{3}}{v}\right)^{3/2}+\cdots\right)\,.\end{split}$
(75)

Here the dependence of $m(v)$ on $Q$ and $l$ are encoded in $m_{cr}$. The
expansion shows that $\lim_{v\rightarrow\infty}m(v)=m_{cr}$, giving the
expected mass for the cold remnant. Remarkably, the late-time behavior is
universal in the sense that there are no free parameters entering the first
three terms of the expansion.

Starting from Price’s law (33), we then include the dynamics of the background
by $m_{0}\mapsto m_{\rm Hawking}(v)$

$m_{-}(v)=m_{\rm Hawking}(v)-\frac{\beta}{(v/v_{0})^{p-1}}\,.$ (76)

The expansion (75) then entails that the late-time behavior of $m_{-}(v)$ is
actually fixed by the Hawking effect, the contribution of the Price tail being
subleading compared to the time-dependence of $m_{\rm Hawking}(v)$. Moreover,
the position of the roots of the lapse function $f(r,v)$ is no longer static:
one inherits an apparent event horizon $r_{+}(v)$ and an apparent Cauchy
horizon $r_{-}(v)$ which both approach the critical radius $r_{cr}$ at
asymptotically late times.

Figure 8: Illustration of the shell dynamics for the background spacetime
given by an evaporating Reissner-Nordström (left) and Hayward black hole
(right) (blue lines). The gray lines depict the positions of the apparent
event (top) and Cauchy horizons (bottom). In both cases the dashed line
depicts $r_{cr}$. The model parameters $Q$ and $l$ have been chosen such that
the critical mass of the remnant is $m_{cr}=1$.

Having $m_{-}(v)$ at our disposal, we can again use (64) to find the dynamics
of the shell. The result obtained from numerical integration is illustrated in
Fig. 8. Here the position of the apparent horizons is indicated by the opaque
gray lines. In analogy to the static case, a shell starting between the
apparent horizons quickly falls towards the apparent Cauchy horizon and
subsequently trails the dynamics of $r_{-}(v)$. This behavior is independent
of the initial conditions and details of the geometry.

The crucial difference with the static background then occurs at the level of
(72). Including the mass loss one has

$\begin{split}{\rm
RN:}\qquad&\,F(v)\simeq-\left(\frac{15\pi\,m_{cr}^{3}}{v}\right)^{1/2}\,,\\\
{\rm
H:}\qquad&\,F(v)\simeq-\left(\frac{80\pi\,m_{cr}^{3}}{v}\right)^{1/2}\,.\end{split}$
(77)

Thus the function $F(v)$ appearing on the right-hand side of the dynamical
equation for $m_{+}(v)$ vanishes asymptotically. The power-law governing the
decay of $F(v)$ is again independent of the geometry under consideration.
Heuristically, this can be understood from the fact that the asymptotic
geometry is a remnant with vanishing surface gravity. Hence the leading terms
in (72) vanish and the dynamics of $F(v)$ starts with the subleading order as
compared to the static case.

Figure 9: Top: Illustration of the mass function $m_{+}(v)$ inside the shell
for a Reissner-Nordström (left) and Hayward black hole (right) undergoing
mass-loss due to Hawking radiation. Initial conditions are imposed at late
times. All green lines are attracted to the late-time attractor where
$m_{+}(v)$ approaches the constant
$m_{+}^{\infty}\equiv\lim_{v\rightarrow\infty}m_{+}(v)$. Botton: Illustration
of the Kretschmann scalar evaluated at the position of the shell. For the
Reissner-Nordström attractor $K|_{\Sigma}$ remains finite. In the Hayward
case, it grows polynomially in $v$. The model parameters $Q$ and $l$ have been
chosen such that $m_{cr}=1$.

The modification (77) has profound consequences for the late-time behavior of
$m_{+}(v)$. In contrast to the static case, the mass function in the interior
of the shell approaches a constant value

$\begin{split}{\rm RN:}\qquad&m_{+}(v)\simeq
m_{cr}+\frac{30\pi\,m_{cr}^{4}}{v}\,,\\\ {\rm
H:}\qquad&m_{+}(v)\simeq-2m_{cr}+12m_{cr}\,\left(\frac{80\pi\,m_{cr}^{3}}{v}\right)^{1/2}\,.\end{split}$
(78)

Again this new attractor appears in both geometries. The different fall-off
behavior in the subleading term for the Reissner-Nordström case is due to the
cancellation of the $v^{-1/2}$-contributions in the series. The approach of
numerical solutions to this late-time attractor is illustrated in Fig. 9, top
row. The new attractor behavior then also propagates into the Kretschmann
scalar evaluated at the position of the shell

$\begin{split}{\rm RN:}\qquad&K|_{\Sigma}\simeq\frac{8}{m_{cr}^{4}}\,,\\\ {\rm
H:}\qquad&K|_{\Sigma}\simeq\frac{59049\,v^{6}}{4096\,m_{cr}^{10}}\,.\end{split}$
(79)

The approach of numerical solutions to this attractor is illustrated in the
bottom row of Fig. 9. The remarkable feature of this result is that _there is
no curvature singularity building up in the Reissner-Nordström case_. The
mechanism underlying the polynomial growths in the Hayward case is identical
to the one observed in the static case. A consistent solution for $m_{+}(v)$
requires the vanishing of the first factor in $p(m_{+})$ which then leads to
cancellations among leading terms in the denominator of the curvature scalar.
As a result, one again experiences a power-law growth of $K|_{\Sigma}$.

An important prerequisite for solutions following the attractor (78) is that
the initial conditions for the shell are imposed at sufficiently late times,
close to the regime where the black hole has almost reached the final phase of
its evaporation process, cf. Fig. 3. In light of the discussion Carballo-
Rubio:2021bpr ; DiFilippo:2022qkl , it is important to clarify whether this
attractor can also be reached from initial conditions imposed at early times,
$v=1$ say. While the corresponding analysis is most likely outside of the
validity range of the Ori-model (the Price tail and the approximation of the
perturbation by a thin shell being reasonable at asymptotically late times
only), we give a tentative answer in Fig. 10. This reveals that the static and
dynamical background models share the same early-time behavior: for the
Reissner-Nordström black hole all solutions again terminate at finite $v$. For
the Hayward case there is again a range of initial conditions where the
solutions connect to the asympotic late-time attractors. These are highlighted
by the green lines in the right diagram of Fig. 10. The existence of these
solutions can again be traced back to the quadratic nature of $p(m_{+})$ which
induces a plateau for $m_{+}$ even before one enters into the regime where
$F(v)$ decays.

Figure 10: Solutions for the mass function $m_{+}$ in the inner sector of the
shell for the Reissner-Nordström (left) and Hayward black hole (right).
Initial conditions are imposed at “early times” $v=1$ and the background is
provided by the most left trajectories $R(v)$ shown in Fig. 8. The blue
solutions terminate at finite value of $v$. For the Hayward case there exists
a critical value for $m_{+}(v)$. Solutions starting above this critical value
are depicted by the green lines and extended up to $v=\infty$. Again $Q$ and
$l$ are chosen such that $m_{cr}=1$.

In order to conclude our discussion, it is important to highlight the
relevance of the attractor (78) with respect to the geodesic completeness of
spacetime. In this context, we stress that the asymptotic geometry is an
extremal black hole. Investigating the geodesics of radially free-falling
massive observers in such a background one finds that the curvature
singularity building up at $v=\infty$ can no longer be reached in a finite
amount of the observer’s proper time. Hence these observers will not encounter
the singularity and questions related to its strength may actually become
academic. Whether this result extends to all observers and the full dynamical
background is currently still an open question. Nevertheless, the exposition
in this section shows that the dynamics of the black hole mass function
induced by the emission of Hawking radiation is a crucial element when
analyzing the late-time stability of regular black holes.

##  6 Conclusions

Static black hole solutions in general relativity are characterized by
curvature singularities hidden behind an event horizon. The singularities are
often taken as a signal for the breakdown of the classical theory which should
be removed by quantum (gravity) effects. A common strategy anticipating such
an effect replaces the singular part of the black hole spacetime by a regular
patch of de Sitter space. In this way one arrives at regular black holes
satisfying the limited curvature hypothesis ansoldi . Examples include
Bardeen-type black holes bardeen1968non ; Ayon-Beato:2000mjt , the Hayward
geometry hayward , RG-improved black holes Bonanno:2000ep  and Planck stars
Rovelli:2014cta .

A direct consequence of the de Sitter core is that asymptotically flat regular
black holes must have (at least) two horizons, an outer event horizon and an
inner Cauchy horizon. Thus, in their simplest incarnation they have the same
horizon structure as a charged Reissner-Nordström black hole. The appearance
of the Cauchy horizon entails drastic consequences for the black hole
evaporation process due to the emission of Hawking radiation: instead of
evaporating completely within a finite time-span, the final state of a
spherically symmetric regular black hole is a cold, regular remnant
corresponding to an extremal black hole.

The similarity to the Reissner-Nordström geometry suggests that regular black
holes may suffer from a dynamical instability, the so-called mass-inflation
effect. In brief the effect states that a tiny perturbation crossing the event
horizon and impacting on the Cauchy horizon induces a curvature singularity in
its interior. Extrapolating this effect from a static, non-extremal Reissner-
Nordström black hole to regular black holes then suggests that the instability
can reintroduce spacetime singularities dynamically.

The analysis of the mass-inflation effect based on the Ori-model reveals that
this analogy comes with significant limitations though Bonanno:2020fgp ;
Bonanno:2022jjp . At the level of static, non-extremal black holes the
equation controlling the dynamics of the mass-inflation effect _is
structurally different_ for the Reissner-Nordström geometry and regular black
holes of the Hayward and RG-improved type. As a consequence, the latter admit
solutions where the mass-function remains constant and the curvature in the
inner sector of the shell grows polynomial in the ingoing Eddington-
Finkelstein coordinate $v$ only.555This conclusion has been challenged in
Carballo-Rubio:2021bpr ; DiFilippo:2022qkl  (also see Carballo-Rubio:2018pmi
for an earlier analysis based on the DTR-relations), claiming that the onset
of the attractor behavior is preceded by a “fatal” phase of exponential
growth. This conclusion is flawed for three reasons though. Firstly, it builds
on an analysis of the system at “early times” outside the region of validity
of the underlying assumptions: Price’s law holds at asymptotically late times
only where the optical geometric limit is valid and one is allowed to neglect
the “finite size” effects of the inner potential barrier. In a realistic
collapse the early-time dynamics is significantly more complicated because of
the presence of the flux from the collapsing star. Secondly, closing eyes to
this difficulty and extrapolating the model to early times, Fig. 7 establishes
that the analysis of Carballo-Rubio:2021bpr ; DiFilippo:2022qkl  is
incomplete. There are initial conditions which actually reach the salient
late-time attractor also from early times. Determining the precise initial
data requires the analysis of the full dynamical process and is beyond the
scope of a simplified model whose fundamental limitation is the assumption
that the Cauchy Horizon always exist at $v=\infty$. Thirdly, a divergence in
the asymptotic mass $m_{+}$ may just not be fatal for the geometry as the
Misner-Sharp mass controlling the curvature of spacetime may remain finite.
This is clear from eq. (65) which shows that the limit $m\rightarrow\infty$
has different physical implications for the Reissner-Nordström and Hayward
geometry. Most remarkably, very similar conclusions have already been reached
when analyzing the mass-inflation effect for loop black holes Brown:2011tv .

The static analysis is modified significantly once the mass-loss due to
Hawking radiation is taken into account Bonanno:2022jjp . This leads to two
novel types late-time attractors where the curvature either grows polynomially
in $v$ (Hayward, RG-improved) or remains finite (Reissner-Nordström, Bardeen).
Moreover, the fact that the final state is an extremal black hole suggests
that no observer can actually reach these singularities in a finite proper
time. These features are important theoretical prerequisits when trying to
establish regular black holes as valid alternatives to the black holes from
general relativity.

The existence of late-time attractors taming or even expelling the mass-
inflation effect for regular black holes is highly encouraging. This raises
the crucial question whether the full dynamics actually reaches these salient
regimes. Settling this question may require a full-fledged numerical analysis
beyond the analytic models describing the mass-inflation effect at late times.
Making this connection will again be an important step towards establishing
regular black holes as valid alternatives to the black holes described by
general relativity.

###### Acknowledgements.

We thank our collaborators N. Alkhofer, J. Daas, M. Galis, G. d’Odorico, I.
van der Pas, A. Platania, A. Khosravi, B. Koch,, S. Silveravalle, F. Vidotto,
M. Wondrak, and, foremost, M. Reuter for many inspiring discussions developing
our understanding of spacetime singularities, regular black holes, and the
stability properties of Cauchy horizons. The work of F.S. is supported by the
Dutch Black Hole Consortium.

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