# Black holes without Cauchy horizons and integrable singularities

J. Ovalle<EMAIL_ADDRESS>Research Centre for Theoretical Physics
and Astrophysics, Institute of Physics, Silesian University in Opava, CZ-746
01 Opava, Czech Republic.

###### Abstract

In general relativity, nonsingular black holes contain (at least) a Cauchy
horizon, a null hypersurface beyond which determinism breaks down. Even though
the strong cosmic censorship conjecture establishes the impossibility of
extending spacetime beyond this region, in this paper we investigate how far
we can go, without invoking this conjecture, in the building of a physically
reasonable black hole without a Cauchy hypersurface. Following this reasoning,
we find a black hole lacking of Cauchy horizon, asymptotically flat and
satisfying either the strong or dominant energy condition. The above is
possible by demanding integrable singularity for the Ricci scalar, whose
direct consequence is the appearance of finite tidal forces. We show that the
spacetime inside the event horizon represents a warped anti-de Sitter
spacetime, which might be interpreted in terms of a finite superposition of
configurations.

##  I Introduction

One of the most notable achievements of general relativity (GR) is the
successful prediction of black holes (BHs) Abbott  _et al._ (2016); Akiyama
_et al._ (2019). It also predicts, under a set of physically reasonable
conditions, the appearance of singularities as final stage of gravitational
collapse, which, as the weak cosmic censorship conjecture (CCC) states, are
hidden inside an event horizon, thus preventing that naked singularities can
be seen by external observers Penrose (1969); Hawking and Ellis (2011).
However, the very existence of singularities, naked or dressed by an event
horizon, should be a clear signal that GR has already reached its extreme
domain, and therefore is falling apart. It is generally believed that such a
problem would be overcome by quantum gravity, even though its complete
formulation is still unknown.

Although we cannot understand the true nature of singularities in the
framework of GR, it is possible to evade them by following a fairly simple
strategy: generate regular BHs by filling the spacetime around the central
singularity with some physically reasonable source of matter. This has
produced a plethora of new regular BH solutions in recent years, mainly
because the matter source used to evade the central singularity can be
interpreted in terms of nonlinear electrodynamics. However, all these regular
BH solutions contain a Cauchy horizon, a null hypersurface beyond which
predictability breaks down, and also leads to mass inflation at the
perturbative level Poisson and Israel (1989, 1990), a pathology which occurs
even in loop quantum gravity inspired models Brown  _et al._ (2011). Regarding
mass inflation, some important advances have been made recently Bonanno  _et
al._ (2021); Carballo-Rubio  _et al._ (2022); Franzin  _et al._ (2022);
Bonanno  _et al._ (2022), showing that singularity regularization does not
always result in exponential instabilities. However, the existence of the
Cauchy horizon remains potentially problematic Carballo-Rubio  _et al._
(2018). A possible solution would be to generate regular BHs without inner
horizon, and indeed some progress has been done in this direction by using
classical Boos (2021) and quantum arguments Casadio  _et al._ (2022, 2023). Of
course, we can always invoke the strong CCC Penrose (1969), which essentially
ensures the impossibility of extending spacetime beyond the Cauchy horizon as
a continuous Lorentzian manifold. But this is precisely contrary to what we
want to develop, i.e., to avoid the very existence of Cauchy horizons.

Regarding the problematic existence of two horizons, there are well-known
particular configurations in GR where the interior and exterior horizon are
merged. These are extremal BHs, whose astrophysical relevance is null, at
least for solutions known so far. Despite this fact, we wonder whether it is
feasible that a regular single-horizon BH could be the final result from
astrophysical collapse of compact objects. The aim of this paper is to
investigate this point, exploring the possibility of building an
astrophysically realistic example of GR being able to cope in high-curvature
domains. In this respect, and even though there is not any rigorous
formulation, it is generally believed that in the framework of classical GR it
is impossible to have a regular BH (i) without a Cauchy horizon, (ii)
asymptotically flat and (iii) satisfying either the strong or dominant energy
condition. So far no counterexample is known, reinforcing the belief that such
configurations cannot exist. In this paper we show that a class of such BHs
does indeed exist by demanding integrable singularity for the Ricci scalar
Lukash and Strokov (2013), whose immediate physical consequence is the
existence of finite tidal forces everywhere, even around the center of the BH
configuration. To accomplish the above, we will build a “Cauchy horizon free”
mass function for a metric describing the interior of a BH. As consequence of
Einstein’s equations, we find that this configuration contains an anti-de
Sitter ($AdS$) spacetime filled with a regular matter function $\hat{m}(r)$,
which is precisely the responsible for eliminating the Cauchy horizon.

##  II Kerr-Schild spacetimes

Let us start from the standard Einstein field equations

$R_{\mu\nu}-\frac{1}{2}\,R\,g_{\mu\nu}=\kappa\,{T}_{\mu\nu}\ ,$ (1)

with $\kappa=8\,\pi\,G_{\rm N}$ and $c=1$. It is well known that the line
element for all spherically symmetric and static spacetimes can be written as
Visser (1995)

$ds^{2}=-e^{\Phi(r)}\left[1-\frac{2m(r)}{r}\right]\,dt^{2}+\frac{dr^{2}}{1-\frac{2m(r)}{r}}+r^{2}d\Omega^{2}\
,$ (2)

where $\Phi(r)$ is a metric function and $m(r)$ stands for the Misner-Sharp
mass function, which measures the amount of energy within a sphere of areal
radius $r$. A particularly interesting case is that where the metric function
$\Phi(r)=0$. Under this condition the line element (2) belongs to the so-
called spacetimes of the Kerr-Schild class Kerr and Schild (1965), which has
been extensively studied (see e.g. Ref. Jacobson (2007)). In this case
Einstein field equations (1) become

$\displaystyle\epsilon=\frac{2{m}^{\prime}}{\kappa\,r^{2}}\
,\,\,\,\,\,\,p_{r}=-\frac{2{m}^{\prime}}{\kappa\,r^{2}}\
,\,\,\,\,\,\,p_{\theta}=-\frac{{m}^{\prime\prime}}{\kappa\,r}\ .$ (3)

where the energy-momentum tensor

$\displaystyle
T_{\mu}^{\,\,\nu}=diag[-\epsilon,\,p_{r},\,p_{\theta},\,p_{\theta}]\ ,$ (4)

contains an energy density $\epsilon$, radial pressure $p_{r}$ and transverse
pressure $p_{\theta}$. Notice a characteristic feature of Einstein equations
in (3), which is a direct consequence of $\Phi(r)=0$, namely, the linearity in
(derivatives of) the mass function ${m}(r)$. Hence, any solution $m(r)$ of the
system (3) can be coupled with a second one $\hat{m}(r)$ to generate a new
solution $\bar{m}(r)$ as

$\displaystyle m(r)$ $\displaystyle\rightarrow$
$\displaystyle\bar{m}(r)=m(r)+\hat{m}(r)\ .$ (5)

The expression in Eq. (5) represents a trivial case of the so-called
gravitational decoupling Ovalle (2017, 2019).

Since the Einstein tensor in Eq. (1) satisfies the contracted Bianchi
identities $\nabla_{\mu}\,G^{\mu}_{\,\,\nu}=0$, the source $T_{\mu\nu}$ must
be covariantly conserved, i.e., $\nabla_{\mu}\,{T}^{\mu\nu}=0$, yielding

$\displaystyle
p_{r}^{\prime}=-\left[\frac{\cancelto{0}{\Phi}^{\prime}}{2}+\frac{m-r\,m^{\prime}}{r(r-2\,m)}\right]\cancelto{0}{\left(\epsilon+p_{r}\right)}\,\,\,\,\,+\,\,\frac{2}{r}\left(p_{\theta}-p_{r}\right)\
.$ (6)

If the metric (2) with $\Phi(r)=0$ describes the interior of a stellar object,
we should expect the density decays monotonically from a maximum at the origin
$r=0$. Hence, from the expressions in Eq. (3), exactly the opposite will occur
for the radial pressure, i.e.,

$\epsilon^{\prime}=-p^{\prime}_{r}<0\rightarrow\,p^{\prime}_{r}>0\ ,$ (7)

and therefore, according to Eq. (6), the interior must be anisotropic, with
the transverse pressure always satisfying $p_{\theta}>p_{r}$. This is
characteristic for any self-gravitating body whose interior is described by a
Kerr-Schild spacetime. In this respect, we want to highlight a critical
feature regarding a Kerr-Schild fluid under hydrostatic equilibrium, which can
be read directly from the expression (6). Let us start by considering an
element of volume $d\tau\,=4\pi\,r^{2}\,dr$ located at $r$ and in equilibrium.
We see that, contrary to what happens in a conventional fluid, there is no
“gravitational force” $\sim\,\left(\epsilon+p_{r}\right)$ acting on the fluid
element $d\,\tau$. Instead, it experiences a pull towards the center as a
consequence of radial pressure gradients $p_{r}^{\prime}>0$, which is canceled
by a push towards the surface, a gravitational repulsion, precisely caused by
the anisotropy in the pressures $\sim\,\left(p_{\theta}-p_{r}\right)$.

Another critical aspect is the continuity of the metric (2) with $\Phi(r)=0$.
If this describes the spacetime of both an inner and outer region of a stellar
object of radii $r_{\rm s}$, then, in order to smoothly joint both regions at
$r=r_{\rm s}$, the mass function $m(r)$ must satisfy

$m(r_{\rm s})=\tilde{m}(r_{\rm s})\ ;\,\,\,\,\,\,\,m^{\prime}(r_{\rm
s})=\tilde{m}^{\prime}(r_{\rm s})\ ,$ (8)

where $\tilde{m}$ stands for the exterior mass function and $F(r_{\rm
s})\equiv\,F(r)\big{\rvert}_{r=r_{\rm s}}$ for any function $F(r)$.
Expressions in Eq. (8) are the necessary and sufficient conditions for
smoothly joint both mass functions $\\{m(r),\,\tilde{m}(r)\\}$ at $r=r_{\rm
s}$. From the expressions in Eqs. (3) and (8) we conclude a significant
feature regarding Kerr-Schild spacetimes: both density and radial pressure are
continuous at the boundary $r=r_{\rm s}$, that is,

$\epsilon(r_{\rm s}^{-})=\epsilon(r_{\rm s}^{+})\ ;\,\,\,\,\,\,p_{r}(r_{\rm
s}^{-})=p_{r}(r_{\rm s}^{+})\ .$ (9)

However, the pressure $p_{\theta}$ is in general discontinuous.

##  III Regularity without
Cauchy horizon

A simple strategy to investigate the existence of regular BHs with a single
horizon is coupling different configurations, as shown in Eq. (5), until
achieving the construction of a convenient mass function, i.e., that where any
potential Cauchy horizon has been removed. However, despite the unlimited
number of couplings we may carry out, when we demand a set of physically
reasonable conditions, the chance of success in such a construction is
extremely limited. Following this strategy, we provide what we consider to be
the simplest possible nonsingular line element, representing a spherically
symmetric ultracompact configuration of radius $r_{\rm s}$, and lacking any
potential Cauchy horizon, namely, the metric (2) with $\Phi(r)=0$ and mass
function

$m^{\pm}(r)=\frac{r}{2}\left[1\pm\left[1-\left(\frac{r}{r_{\rm
s}}\right)^{n}\right]^{k}\right]\ ;\;\;\;\;\;\ r\leq r_{\rm s}$ (10)

where $\\{k,\,n\\}$ are constants to be discussed later. A simple analysis of
the causal structure of this metric shows that it represents a BH for
$m^{+}(r)$ and a horizonless ultracompact configuration for $m^{-}(r)$. In
this work we will analyze the BH solution, leaving the horizonless case for
future investigation. At this point we wish to stress that we do not make any
attempt to postulate any action from which to derive the metric (2) with
$\\{\Phi=0,\,m^{\pm}\\}$. Quite the contrary, our goal is to prove the
existence of BH without Cauchy horizons, being simultaneously more tolerable
to the singularity problem. In this respect, we remark the metric generated by
the mass function (10) is not globally defined but confined to $r\leq\,r_{\rm
s}$, where the total mass $M$ of the system, defined as

$M\equiv\,m(r_{\rm s})=\frac{r_{\rm s}}{2}\ ,$ (11)

is independent of $\\{k,\,n\\}$, and therefore the event horizon is equal to
the Schwarzschild’s radius $r_{\rm s}=2\,M$. The source
$\\{\epsilon,\,p_{r},\,p_{\theta}\\}$ generating the BH solution, i.e., the
metric (2) with $\\{\Phi=0,\,m^{+}\\}$, is found by using the mass function
$m^{+}$ in Eq. (3). A simple analysis shows that a monotonic decrease of the
density $\epsilon(r)$ with increasing radius is only possible for i) $k=1;$
$n\in[\,0,1\,)$ and ii) $k>1;$ $n\in(\,1,2\,]$. In this regard, even though we
should expect violation of energy conditions Martin-Moruno and Visser (2017),
we found that the strong ($k=1$) and dominant ($k>6$) are satisfied. For the
strong energy conditions, we have

$\displaystyle\cancelto{0}{{\epsilon}+{p}_{r}}+\,{p}_{\theta}=\frac{n(n+1)}{2\,r^{2}}\left(\frac{r}{r_{\rm
s}}\right)^{n}\geq 0\ ,$ (12)
$\displaystyle{\epsilon}+{p}_{\theta}=\frac{4+(n+1)(n-2)\left(\frac{r}{r_{\rm
s}}\right)^{n}}{2\,r^{2}}\geq 0\ ,$ (13)

where the expressions in Eqs. (3) and (10) have been used. We can proceed in a
similar way (with arbitrary $k$) for the dominant energy conditions,
$\epsilon\,\geq\,0\ ;\epsilon\,\geq\,|{p}_{i}|\quad\left(i=r,\theta\right)$,
although the expressions are more involved.

We see that the parameters $\\{k,\,n\\}$ produce a rich and physically
attractive setting well worth investigating. However, keeping in mind the
importance of vacuum energy as a source for quantum effects, and the well-
known existence of a de Sitter core in regular BHs, in this work we will only
be interested in the case $n=2$, whose physical interpretation is quite
straightforward, as we will see below. First of all, notice that the case
$\\{k=1,\,n=2\\}$ yields

$R=-4\,\Lambda+4/r^{2}\ ,$ (14)

where $R$ and $\Lambda=3/r_{\rm s}^{2}$ are respectively the scalar curvature
and cosmological constant. The expression in Eq. (14) corresponds to an anti-
de Sitter ($AdS$) spacetime filled with some matter producing a scalar
singularity at the origin. Following the expressions in Eqs. (5) and (10), we
find that the $AdS$ behind our solution was deformed by coupling $m_{\rm
AdS}\rightarrow\,m_{\rm AdS}+\hat{m}$, where $m_{\rm AdS}$ and $\hat{m}$ are
respectively the $AdS$ mass function and $\hat{m}=r$. We see that the mass
function $\hat{m}$ producing the singularity in Eq. (14) is perfectly regular.
We conclude that the singularity displayed in Eq. (14), necessary to remove
the Cauchy horizon, is indeed integrable, where tidal forces remain finite
Lukash and Strokov (2013) [notice that $\int\,R\,d\tau={\rm finite}$ yields
$\int\,\xi_{i}\,d\tau={\rm finite}$ for each diagonal element $\xi_{i}$ in Eq.
(4), since $R\,\sim\,T$]. Now we can safely say that $k>1$ measures deviations
from the deformed $AdS$ in Eq. (14). Indeed, by simple expansion of the metric
function, i.e.

$+g_{tt}=1-k\left(\frac{r}{r_{\rm
s}}\right)^{2}+\underbrace{\frac{k(k-1)}{2!}\left(\frac{r}{r_{\rm
s}}\right)^{4}{+\,.\,.\,.}}_{\rm``deformation^{\prime\prime}}$ (15)

we can identify the effective cosmological constant as
$\Lambda_{k}=3\,k/r_{\rm s}^{2}$. Finally, we want to highlight a critical
point regarding the role of $k$ as generator of the deformed $AdS$. Notice
that so far $k>1$ is an arbitrary constant, namely, $k\,\in\mathbb{R}$.
However, when we impose $k\,\in\mathbb{N}$, the warped $AdS$ has a simple and
fairly straightforward interpretation [see Eqs. (5) and (10)]: it is a finite
superposition of configurations. In particular, we can write the energy
density in Eq. (3) as

$\displaystyle\kappa\epsilon_{k}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}+k!\sum_{n=1}^{k}(-1)^{n}\frac{2n+1}{n!(k-n)!}\frac{r^{2(n-1)}}{r_{\rm
s}^{2n}}$ (16) $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}+k!(-\Lambda+\epsilon^{(2)}_{k}-...+\epsilon^{(k)}_{k})\
,$

showing a finite superposition of $k$ configurations $\epsilon^{(n)}_{k}$,
where $\epsilon_{\rm AdS}\equiv\epsilon_{1}$ is the (negative) vacuum energy,
namely, the cosmological constant $\Lambda=3/r_{\rm s}^{2}$. Since some
configurations $\epsilon^{(n)}_{k}$ are negatives, we can interpret (16) as a
superposition of fluctuations in the $\epsilon_{\rm AdS}$ vacuum state,
something which is quite suggestive as reminiscent of quantum fluctuations.

Clearly the solution generated by the mass function (10) is not asymptotically
flat. In order to build a proper solution globally defined, we start by
exploring whether is possible a smooth transition at $r=r_{\rm s}$ between the
interior solution, given by the metric (2) with $\\{\Phi=0,\,m^{+}\\}$, and an
exterior with $\\{\Phi=0,\,\tilde{m}(r)\\}$, where $\tilde{m}(r)$ is the mass
function for $r\geq r_{\rm s}$. First of all, notice that the interior radial
pressure in Eq. (3) at the surface $r=r_{\rm s}^{-}$ is given by

$p_{r}(r_{\rm s}^{-})=-\frac{1}{\kappa\,r_{\rm s}^{2}}\neq\,0\ .$ (17)

Hence, according to the matching condition (9), we conclude that the metric
(2) with $\\{\Phi=0,\,m^{+}\\}$ cannot be smoothly coupled with the
Schwarzschild’s exterior solution (vacuum)

$ds^{2}=-\left(1-\frac{2{\cal M}}{r}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2{\cal
M}}{r}}+r^{2}d\Omega^{2}\ .$ (18)

Therefore, we need a new exterior solution for the immediate surrounding
region $r\sim r_{\rm s}^{+}$ which quickly goes to Schwarzschild’s exterior.
This will be a transient region between the metrics (2) with
$\\{\Phi=0,\,m^{+}\\}$ and (18). To accomplish this, we will make use of a new
exterior solution recently developed through the so-called gravitational
decoupling Ovalle  _et al._ (2020), namely, a “tensor-vacuum” (an analogy to
scalar vacuum) which satisfies the strong energy condition, and whose mass
function reads

$\tilde{m}(r)={\cal M}-\alpha\frac{r}{2}\,e^{-2r/(2\mathcal{M}-\ell)}\ ,$ (19)

where $\mathcal{M}$ is the asymptotic mass (the total mass measure by an
asymptotic observer), $\alpha$ measures deviations from Schwarzschild’s
solution, and $\ell$ a “hair” which for now satisfies $0<\ell\leq 2\,{\cal M}$
to ensure asymptotic flatness, and whose physical meaning will be explained
below. At this stage it is important to emphasize the difference between
${\cal M}$ and $M$ in Eq. (11). The former is the total mass of the
configuration, and in general is extended beyond $r=r_{\rm s}$, while the
latter is the fraction of this total mass confined within the region $r\leq
r_{\rm s}$.

Using the mass functions (10) and (19) in the matching conditions (8), we find
$\\{\mathcal{M},\,\alpha\\}$ in terms of $\ell$ as

$\displaystyle 4{\cal M}_{\pm}$ $\displaystyle=$ $\displaystyle 2r_{\rm
s}+\ell\pm\sqrt{\ell^{2}+4r_{\rm s}\,\ell-4r_{\rm s}^{2}}\ ,$ (20)
$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\left(\frac{{2\cal
M}}{r_{\rm s}}-1\right)e^{\frac{2r_{\rm s}}{2\cal M-\ell}}\ ,$ (21)

where $\ell\geq\,2r_{\rm s}(\sqrt{2}-1)$. By using the matching condition
(21), we find the exterior mass function (19) as

$\tilde{m}(r)={\cal M}+\frac{r}{2}\left(1-\frac{{2\cal M}}{r_{\rm
s}}\right)e^{-2(r-r_{\rm s})/(2\mathcal{M}-\ell)}\ ,$ (22)

where the mass ${\cal M}$ in Eq. (20) has two branches $\\{{\cal
M}_{+},\,{\cal M}_{-}\\}$ covering, respectively, ${\cal M}\geq\sqrt{2}r_{\rm
s}/2$ and ${\cal M}\leq\sqrt{2}r_{\rm s}/2$. Plugging the mass function (22)
into the line element (2) with $\Phi(r)=0$, we obtain the metric for the
region $r\geq r_{\rm s}$ as

$\displaystyle ds^{2}=-$ $\displaystyle\left[1-\frac{{2\cal
M}}{r}-\left(1-\frac{{2\cal M}}{r_{\rm s}}\right)e^{-2(r-r_{\rm
s})/(2\mathcal{M}-\ell)}\right]dt^{2}$ $\displaystyle+$
$\displaystyle\left[1-\frac{{2\cal M}}{r}-\left(1-\frac{{2\cal M}}{r_{\rm
s}}\right)e^{-2(r-r_{\rm s})/(2\mathcal{M}-\ell)}\right]^{-1}dr^{2}$
$\displaystyle+$ $\displaystyle r^{2}d\Omega^{2}\ .$ (23)

By simple inspection of the metric (23), we see that it represents a black
hole with horizon

$r_{\rm s}=2\,M\,=\,({\cal M}+\sqrt{\ell{\cal M}-{\cal M}^{2}})\ ,$ (24)

and primary hair $\ell$ satisfying ${\cal M}\leq\ell\leq 2\,{\cal M}$. In
order to elucidate the physical meaning of this parameter, notice that from
Eq. (24) we find

$\displaystyle\ell\rightarrow\,2{\cal M}\,\,\Rightarrow\,{\cal
M}\rightarrow\,M\ .$ (25)

Hence, we can say that $\ell$ controls the amount of mass ${\cal M}$ contained
within the trapped surface $r=r_{\rm s}$. We see that $\ell=2{\cal M}$ in Eq.
(25) represents a Schwarzschild’s BH, where the total mass ${\cal M}$ is
confined within the region $r\leq r_{\rm s}$. However, this case is explicitly
excluded by the matching condition (9), and therefore ${\cal
M}\leq\ell<2\,{\cal M}$. Nevertheless, the case $\ell\sim\,2{\cal M}$, i.e.,
${\cal M}\sim\,M$ deserve more attention, as we will discuss later. We
conclude by specifying the mass function

$m(r)=\left\\{\begin{array}[]{l}\frac{r}{2}\left[1+\left[1-\left(\frac{r}{r_{\rm
s}}\right)^{2}\right]^{k}\right]\ ;\,\,\,\,\,\,\,\,0\leq\,r\leq r_{\rm s}\\\
\\\ {\cal M}+\frac{r}{2}\left(1-\frac{{2\cal M}}{r_{\rm
s}}\right)e^{\frac{-2(r-r_{\rm s})}{(2\mathcal{M}-\ell)}}\ ;\,\,\,\,\,\,r\geq
r_{\rm s}\ ,\end{array}\right.$ (26)

which through (2) with $\Phi(r)=0$ yields a BH lacking of Cauchy horizon. We
emphasize this solution contains three parameters $\\{{\cal M},\,k,\,\ell\\}$,
where $k$ drives discrete deformations undergone by the interior $AdS$, and
$\ell$ a primary hair driving the matter dressing the horizon. As expected,
its surface gravity $\kappa$ and Hawking temperature $T_{H}$ are zero. We may
be tempted to ensure that the solution represents a cold remnant. However, the
configuration extends beyond the horizon, where the aforementioned variables
have finite values for $r\sim\,r_{\rm s}^{+}$. Since we have a hairy solution,
we need to specify the Lagrangian ${\cal L}$ associate with the primary hair
$\ell$ to find the entropy $S$ Wald (1993); Visser (1993). In any case, notice
that

$\frac{A_{\rm H}}{4}=\pi\,{\cal M}\left(\ell+2\sqrt{\ell{\cal M}-{\cal
M}^{2}}\right)\ .$ (27)

##  IV Apparent singularity
and final remarks

In order to discuss an apparent nonintegrable singularity at the event
horizon, let us analyze in detail what happens on the surface $r=r_{\rm
s}^{+}$. Since in general $m^{\prime\prime}$ is not continuous, there will be
a discontinuity in the transverse pressure [see Eq. (3)], whose expression for
the outer region is given by

$p^{+}_{\theta}(r)=\frac{2(r+\ell-2{\cal M})({\cal
M}-M)}{\kappa\,M\,r\,(2\mathcal{M}-\ell)^{2}}\,e^{\frac{-2(r-r_{\rm
s})}{(2\mathcal{M}-\ell)}}\ ,$ (28)

We see that $p^{+}_{\theta}(r_{\rm s})\sim\,\frac{1}{2{\cal M}-\ell}$ for
$\ell\rightarrow\,2{\cal M}\,\,({\cal M}\rightarrow\,M)$. We conclude the
greater the mass ${\cal M}$ packed in the region $r\leq r_{\rm s}$, the
greater the transverse stresses on the surface $r=r_{\rm s}$, and indeed it
diverges for ${\cal M}=M$ (which is excluded from our solution). Similarly, we
find that curvature invariants on the horizon behave as

$\displaystyle R\sim\frac{1}{{\cal M}}\frac{1}{2{\cal M}-\ell}\
;\,\,\,\,R_{\mu\nu}R^{\mu\nu}\sim\frac{1}{{\cal M}^{2}}\frac{1}{(2{\cal
M}-\ell)^{2}}\ ;$ (29) $\displaystyle
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\sim\frac{1}{{\cal
M}^{2}}\frac{1}{(2{\cal M}-\ell)^{2}}\ .$ (30)

Although the case $\ell=2{\cal M}$ is excluded from (26), it is worth
investigating a possible solution without $(2{\cal M}-\ell)^{-1}$ in the
curvature invariants. This is accomplished by demanding continuity of
$m^{\prime\prime}$, which leads to $\ell={\cal M}$. In this case we end up
with a simpler BH with only two parameters, i.e., $\\{{\cal M},\,k\\}$, where
$\ell$ gives way to a secondary hair that we can read directly from Eq. (26).
However, only half of the total mass ${\cal M}$ will be enclosed by the
horizon [see Eq (24)], with the other half filling the region $r>r_{\rm s}$,
something whose phenomenological feasibility could mean quite a challenge.
Therefore, in order to control the amount of mass contained within $r=r_{\rm
s}$, we need to keep alive the primary hair $\ell$.

Coming back to our solution (26), the expressions in Eqs. (29) and (30)
clearly indicate that the horizon $r_{\rm s}=2\,M$ must be dressed with an
amount of matter $d\,{\cal M}={\cal M}-M\neq\,0$ that cannot cross $r=r_{\rm
s}$, otherwise a naked singularity arises. This process occurs thanks to the
gravitational repulsion [see Eq. (6)] and seems to indicate a sort of
exclusion mechanism to prevent singularities and other pathologies arising in
classical GR. On the other hand, a phenomenologically viable BH should be that
where almost all of ${\cal M}$ in the metric (23) is contained in the region
$r\leq\,r_{\rm s}$, i.e., $d\,{\cal M}={\cal M}-M\sim\,0$, but this seems to
bring us too close to the potential singularity displayed in Eqs. (29) and
(30) . This leads us to wonder whether it is possible to dress the black hole
with a phenomenologically feasible amount of matter $d\,{\cal M}$ while being
simultaneously far enough from the singularity. A simple analysis shows a
reasonable behavior of the solution for the case ${\cal M}/M\sim\,1.01$, i.e.,
where most of the total mass ${\cal M}$ ($\sim\,99\%$) is contained inside the
trapped surface $r=r_{\rm s}$. Such a small amount of matter $d\,{\cal
M}\sim\,1\%$ is consistent with the fact that a perfect vacuum surrounding a
BH is nothing but an idealized physical setting. Also notice that the more
massive the object, the more tolerance it has for the potential divergence
displayed in Eqs. (29) and (30). Hence, supermassive BHs only need a fairly
small fraction of matter covering their horizon. Finally, some aspects of the
solution should be analyzed in depth, such as its stability, possible
rotational extension and observational consequences, as well as time-dependent
formation and evaporation, and most importantly, the action generating this
solution. In this regard, an important guide should be the anisotropic effects
causing the gravitational repulsion, necessary for noncollapsing matter, and
critical for the existence of finite tidal forces. However, those are quite
another matter beyond the target of this paper.

### Acknowledgments

This work is partially supported by ANID FONDECYT Grant No. 1210041.

## References

  * Abbott  _et al._ (2016) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc] .
  * Akiyama  _et al._ (2019) K. Akiyama _et al._ (Event Horizon Telescope), Astrophys. J. Lett. 875, L1 (2019), arXiv:1906.11238 [astro-ph.GA] .
  * Penrose (1969) R. Penrose, Riv. Nuovo Cim. 1, 252 (1969).
  * Hawking and Ellis (2011) S. W. Hawking and G. F. R. Ellis,  _The Large Scale Structure of Space-Time_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2011).
  * Poisson and Israel (1989) E. Poisson and W. Israel, Phys. Rev. Lett. 63, 1663 (1989).
  * Poisson and Israel (1990) E. Poisson and W. Israel, Phys. Rev. D 41, 1796 (1990).
  * Brown  _et al._ (2011) E. G. Brown, R. B. Mann, and L. Modesto, Phys. Rev. D 84, 104041 (2011), arXiv:1104.3126 [gr-qc] .
  * Bonanno  _et al._ (2021) A. Bonanno, A.-P. Khosravi, and F. Saueressig, Phys. Rev. D 103, 124027 (2021), arXiv:2010.04226 [gr-qc] .
  * Carballo-Rubio  _et al._ (2022) R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, and M. Visser, JHEP 09, 118 (2022), arXiv:2205.13556 [gr-qc] .
  * Franzin  _et al._ (2022) E. Franzin, S. Liberati, J. Mazza, and V. Vellucci, (2022), arXiv:2207.08864 [gr-qc] .
  * Bonanno  _et al._ (2022) A. Bonanno, A.-P. Khosravi, and F. Saueressig, (2022), arXiv:2209.10612 [gr-qc] .
  * Carballo-Rubio  _et al._ (2018) R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, and M. Visser, JHEP 07, 023 (2018), arXiv:1805.02675 [gr-qc] .
  * Boos (2021) J. Boos, (2021), arXiv:2104.00555 [gr-qc] .
  * Casadio  _et al._ (2022) R. Casadio, A. Giusti, and J. Ovalle, Phys. Rev. D 105, 124026 (2022), arXiv:2203.03252 [gr-qc] .
  * Casadio  _et al._ (2023) R. Casadio, A. Giusti, and J. Ovalle, (2023), arXiv:2303.02713 [gr-qc] .
  * Lukash and Strokov (2013) V. N. Lukash and V. N. Strokov, Int. J. Mod. Phys. A 28, 1350007 (2013), arXiv:1301.5544 [gr-qc] .
  * Visser (1995) M. Visser,  _Lorentzian wormholes: From Einstein to Hawking_ (1995).
  * Kerr and Schild (1965) R. P. Kerr and A. Schild, Proc. Symp. Appl. Math 17, 199 (1965).
  * Jacobson (2007) T. Jacobson, Class. Quant. Grav. 24, 5717 (2007), arXiv:0707.3222 [gr-qc] .
  * Ovalle (2017) J. Ovalle, Phys. Rev. D95, 104019 (2017), arXiv:1704.05899 [gr-qc] .
  * Ovalle (2019) J. Ovalle, Phys. Lett. B788, 213 (2019), arXiv:1812.03000 [gr-qc] .
  * Martin-Moruno and Visser (2017) P. Martin-Moruno and M. Visser, Fundam. Theor. Phys. 189, 193 (2017), arXiv:1702.05915 [gr-qc] .
  * Ovalle  _et al._ (2020) J. Ovalle, R. Casadio, E. Contreras, and A. Sotomayor,  (2020), 10.1016/j.dark.2020.100744, arXiv:2006.06735 [gr-qc] .
  * Wald (1993) R. M. Wald, Phys. Rev. D 48, R3427 (1993), arXiv:gr-qc/9307038 .
  * Visser (1993) M. Visser, Phys. Rev. D 48, 5697 (1993), arXiv:hep-th/9307194 .