# Properties of Scalar Hairy Black Holes and Scalarons with Asymmetric
Potential

Xiao Yan Chew<EMAIL_ADDRESS>Department of Physics Education, Pusan
National University, Busan 46241, Republic of Korea  Research Center for
Dielectric and Advanced Matter Physics, Pusan National University, Busan
46241, Republic of Korea  Dong-han Yeom<EMAIL_ADDRESS>Department of
Physics Education, Pusan National University, Busan 46241, Republic of Korea
Research Center for Dielectric and Advanced Matter Physics, Pusan National
University, Busan 46241, Republic of Korea  Jose Luis Blázquez-Salcedo
<EMAIL_ADDRESS>Departamento de Fśica Teórica and IPARCOS, Universidad
Complutense de Madrid, E-28040 Madrid, Spain

###### Abstract

In this paper we study the properties of black holes and scalarons in Einstein
gravity when it is minimally coupled to a scalar field $\phi$ with an
asymmetric potential $V(\phi)$, constructed in [Phys. Rev. D 73 (2006),
084002] a few decades ago. $V(\phi)$ has been applied in the cosmology to
describe the quantum tunneling process from the false vacuum to the true
vacuum and contains a local maximum, a local minimum (false vacuum) and a
global minimum (true vacuum). In particular we focus on the asymptotically
flat solutions, which can be constructed by fixing appropriately the local
minimum of $V$. A branch of hairy black holes solutions emerge from the
Schwarzschild black hole, and we study the domain of existence of such
configurations. They can reach to a particle-like solution in the small
horizon limit, i.e. the scalarons. We study the stability of black holes and
scalarons, showing that all of them are unstable under radial perturbations.

##  I Introduction

The “No hair” theorem states that the properties of black holes are only
described by the mass, angular momentum and electrical charge after
gravitational collapse or any dynamical perturbations of black holes, since
they approach the stationary limit. However, the “No hair” theorem can be
circumvented under the right conditions. For example, the existence of
particle-like solution for SU(2) Einstein-Yang-Mills theory shown by Bartnik
and Mckinnon Bartnik:1988am  had led to the construction of non-Abelian hairy
black holes Bizon:1990sr ; Volkov:1990sva ; Kuenzle:1990is ; Greene:1992fw ;
Lavrelashvili:1992ia  which don’t obey the “No hair theorem” anymore. One of
the simplest way to circumvent the “No hair” theorem is to minimally couple
Einstein gravity with a scalar field, introducing a scalar potential which is
not strictly positive such that the weak energy condition is violated
Herdeiro:2015waa . In Corichi:2005pa  spherically symmetric and asymptotically
flat hairy black holes were constructed by employing a scalar potential which
has a global minimum, a local minimum and a local maximum (asymmetric
potential). They obtained the asymptotically flat black holes by fixing the
local minimum of potential to zero, to mainly study the empirical mass formula
of such black holes Corichi:2005pa  and later also generalize their model to
non-minimally coupled scalar field with gravity Nucamendi:1995ex . However,
the properties of black holes such as the Hawking temperature and mass had not
been investigated systematically in terms of the parameters of the scalar
potential. In this work we investigate the properties of these solutions by
fixing the global minimum of the potential and varying the local maximum. This
allows us to generate a branch of hairy black holes that bifurcate from the
Schwarzschild black holes. See Bechmann:1995sa ; Dennhardt:1996cz ;
Bronnikov:2001ah ; Martinez:2004nb ; Winstanley:2005fu ; Nikonov:2008zz ;
Anabalon:2012ih ; Gao:2021ubl ; Karakasis:2021rpn  for similar constructions
of scalar hairy black holes.

When there is a scalar field with an asymmetric potential, a quantum tunneling
from a false vacuum (a local minimum) to a true vacuum (a global minimum) is
allowed. With an $O(4)$-symmetric metric ansatz, the Coleman-De Luccia
instantons explain such a tunneling process via nucleation of a bubble
Coleman:1980aw . Even we extend to the spherical symmetry, again, one can
build a solution that explains a tunneling process Masoumi:2012yy . For both
cases, inside is a true-vacuum region, and after the nucleation, the bubble
should expand over the spacetime; otherwise, the scalar field combination is,
in general, unstable. Such a bubble may explain the phase transition of the
early universe cosmology DeLuca:2021mlh ; Perivolaropoulos:2022txg ;
Braden:2022odm ; Ekstedt:2022ceo ; Cruz:2022ext ; Vicentini:2022pra ; also,
some interactions between bubbles may be a source of gravitational waves
Kosowsky:1991ua ; Kim:2014ara ; Lee:2021nwg .

On the other hand, one can also say that the same solution can be interpreted
as a kind of (unstable) scalarons or stationary hairy black hole solutions. In
order to provide a smooth field configuration at the horizon over the
Euclidean manifold, one needs to provide the Dirichlet boundary condition at
the event horizon. If we generalize this boundary condition and focus only on
the astrophysical aspects, it is still allowed to provide more generic
boundary conditions than the pure Dirichlet boundary condition Masoumi:2012yy
; Gregory:2020cvy . This is the case that we will investigate in the present
paper.

This paper is organized as follows. In Sec. II, we briefly introduce our
theoretical setup comprising the Lagrangian and the metric ansatz. Then, we
derive the set of coupled differential equations and study the asymptotic
behavior of the functions. In Sec. III, we briefly introduce the quantities of
interest for the black holes. In Sec. IV, we study the stability of the hairy
black holes and scalarons by calculating the unstable mode of the radial
perturbations of the metric and the scalar field. In Sec V, we present and
discuss our numerical findings. Finally, in Sec. VI, we summarize our work and
present an outlook.

##  II Theoretical Setting

###  II.1 Theory and Ansätze

Figure 1: An illustration of a generic asymmetric scalar potential $V(\phi)$.

The action for Einstein gravity minimally couples with an asymmetric potential
$V(\phi)$ of a scalar field $\phi$ is given by Corichi:2005pa

$S=\int d^{4}x\sqrt{-g}\left[\frac{R}{16\pi
G}-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi-V(\phi)\right]\,,$ (1)

where

$V(\phi)=\frac{V_{0}}{12}\left(\phi-a\right)^{2}\left[3\left(\phi-a\right)^{2}-4(\phi-a)(\phi_{0}+\phi_{1})+6\phi_{0}\phi_{1}\right]\,,$
(2)

with $a$, $V_{0}$, $\phi_{0}$ and $\phi_{1}$ are the constants. As shown in
Fig 1, the appearance of cubic term $\phi^{3}$ causes the potential to take
the asymmetric shape. If the cubic term in the potential disappears, then the
potential is Higgs like with two degenerate minima. Here the constant $a$ is
the local minimum of potential, $\phi_{0}$ is the local maximum of potential
and $\phi_{1}$ is the global minimum of potential. Note that
$0<2\phi_{0}<\phi_{1}$. In cosmology, this potential can be used to explain a
quantum tunneling process from the false vacuum $a$ to the true vacuum
$\phi_{1}$. In this paper, we choose $a=0$ such that the hairy black holes is
asymptotically flat at the spatial infinity. However, one could obtain a
Schwarzschild-AdS like solutions if $a$ is non-zero Corichi:2005pa .

The variation of the action with respect to the metric and scalar field yields
the Einstein equation and Klein-Gordon (KG) equation, respectively

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi
GT_{\mu\nu}\,,\quad\nabla_{\mu}\nabla^{\mu}\phi=\frac{dV(\phi)}{d\phi}\,,$ (3)

where the stress-energy tensor $T_{\mu\nu}$ is given by

$T_{\mu\nu}=-g_{\mu\nu}\left(\frac{1}{2}\nabla_{\alpha}\phi\nabla^{\alpha}\phi+V(\phi)\right)+\nabla_{\mu}\phi\nabla_{\nu}\phi\,.$
(4)

We employ the following spherically symmetric Ansatz to construct the
particle-like and black holes solutions,

$ds^{2}=-N(r)e^{-2\sigma(r)}dt^{2}+\frac{dr^{2}}{N(r)}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\,,$ (5)

where $N(r)=1-2m(r)/r$ with $m(r)$ is the Misner-Sharp mass function. Note
that $m(\infty)=M$, the total mass of the configuration.

###  II.2 Justification on the Existence of Hairy Black Holes

Figure 2: Left: The maximally extended causal structure for hairy black holes.
The gray-colored part denotes a region for negative vacuum energy. The black
dashed curve denotes a cusp surface of the scalar field and the blue dashed
curve denotes a star surface. Right: The physically sensible interpretation of
the hairy black hole does not include any cusp region.

Here we briefly justify the existence of hairy black holes by referring to
Herdeiro:2015waa . We multiply $\phi$ to the Klein-Gordon equation and
integrate it from the black hole horizon to infinity,

$\int_{r_{H}}^{\infty}d^{4}x\sqrt{-g}\left[\phi\nabla_{\mu}\nabla^{\mu}\phi-\phi\frac{dV(\phi)}{d\phi}\right]=0\,.$
(6)

We integrate the first term in above expression by parts and obtain,

$\int_{r_{H}}^{\infty}d^{4}x\sqrt{-g}\left[-\nabla_{\mu}\phi\nabla^{\mu}\phi-\phi\frac{dV(\phi)}{d\phi}\right]+\int_{\mathscr{H}}d^{3}\sigma
n^{\mu}\phi\nabla_{\mu}\phi=0\,,$ (7)

where $n^{\mu}$ is the normal vector on the Killing horizon. The second term
in the above expression is the boundary term which vanishes when we apply the
boundary conditions for the scalar field at the horizon with
$n^{\mu}\nabla_{\mu}\phi=0$ and demand the scalar field falls off at infinity.
Hence, we are left with

$\int_{r_{H}}^{\infty}d^{4}x\sqrt{-g}\left[\nabla_{\mu}\phi\nabla^{\mu}\phi+\phi\frac{dV}{d\phi}\right]=0\,,$
(8)

Here $\nabla_{\mu}\phi\nabla^{\mu}\phi\geq 0$ because the gradient of $\phi$
is perpendicular to both Killing vectors and thus has to be spacelike or zero.
Then in order to obtain a regular and nontrivial hairy black hole, the term
$\phi\frac{dV}{d\phi}\leq 0$. In our case, we choose $\phi$ to be always
greater than zero, while the potential $V(\phi)$ is negative in some region;
therefore we allow for the existence of nontrivial scalar hairy black holes.

Furthermore, one can multiply the Klein-Gordon equation by $\frac{dV}{d\phi}$
and repeat again the above procedure, obtaining

$\int_{r_{H}}^{\infty}d^{4}x\sqrt{-g}\left[\frac{d^{2}V}{d\phi^{2}}\nabla_{\mu}\phi\nabla^{\mu}\phi+\left(\frac{dV}{d\phi}\right)^{2}\right]=0\,.$
(9)

In order to make the terms in the square bracket to vanish non-trivially, it
is clear that $\frac{d^{2}V}{d\phi^{2}}<0$, a condition that is also satisfied
in our case. Note that in this derivation, it is not necessary to use the
Einstein equation.

On the other hand, we can also see that the weak energy condition can be
violated since $V$ possesses a global minimum with $V<0$ in some regions of
$\phi$,

$\rho=-T^{t}_{t}=\frac{N}{2}\phi^{\prime 2}+V\,.$ (10)

The violation of weak energy condition leads to the violation of the strong
energy condition (the opposite not being necessarily true). Moreover, we could
also use the Virial identity to reach the same conclusion, that is, $V\leq 0$
in some region, but for this analysis it is necessary to introduce the metric
Ansatz into the action.

(a)  (b)

Figure 3: The properties of hairy black holes $(r_{H}=1,V_{0}=1)$ with several
global minimum $\phi_{1}$ as a function of $\phi_{H}$: (a) The scaled area of
horizon $a_{H}$. (b) The scaled Hawking temperature $t_{H}$.

In summary, this potential $V$ possesses a local maximum and a global minimum
which causes $V$ to be negative in some region. The local minimum at zero
guarantees that the black hole is asymptotically flat. Moreover, it is
possible to repeat the analyisis to show the existence of solitonic solutions,
just by changing the lower end of integration (no horizon with a fully regular
spacetime).

###  II.3 Ordinary Differential Equations (ODEs)

By substituting Eq. (5) into the Einstein-matter field equation, we obtain a
set of second-order and nonlinear ODEs for the metric functions,

$m^{\prime}=2\pi Gr^{2}\left(N\phi^{\prime
2}+2V\right)\,,\quad\sigma^{\prime}=-4\pi Gr\phi^{\prime
2}\,,\quad\left(e^{-\sigma}r^{2}N\phi^{\prime}\right)^{\prime}=e^{-\sigma}r^{2}\frac{dV}{d\phi}\,,$
(11)

where the prime denotes the derivative of the functions with respect to the
radial coordinate $r$.

To construct globally regular black hole solutions, we need to know the
asymptotic behavour of the functions at the horizon and the infinity. By
making the series expansion for the functions at the horizon, the leading
terms in the series expansion are given by

$\displaystyle m(r)$
$\displaystyle=\frac{r_{H}}{2}+m_{1}(r-r_{H})+O\left((r-r_{H})^{2}\right)\,,$
(12) $\displaystyle\sigma(r)$
$\displaystyle=\sigma_{H}+\sigma_{1}(r-r_{H})+O\left((r-r_{H})^{2}\right)\,,$
(13) $\displaystyle\phi(r)$
$\displaystyle=\phi_{H}+\phi_{H,1}(r-r_{H})+O\left((r-r_{H})^{2}\right)\,,$
(14)

where

$m_{1}=4\pi Gr^{2}_{H}V(\phi_{H})\,,\quad\sigma_{1}=-4\pi
Gr_{H}\phi_{1}\,,\quad\phi_{H,1}=\frac{r_{H}\frac{dV(\phi_{H})}{d\phi}}{1-8\pi
Gr_{H}^{2}V(\phi_{H})}\,.$ (15)

Here $\sigma_{H}$ and $\phi_{H}$ are the values of $\sigma$ and $\phi$ at the
horizon. Similarly, the leading terms in the series expansion for scalaron at
the origin are given by

$\displaystyle m(r)$ $\displaystyle=\frac{4\pi
GV(\phi_{c})}{3}r^{3}+O(r^{5})\,,$ (16) $\displaystyle\sigma(r)$
$\displaystyle=\sigma_{c}-\frac{\pi
G}{9}\left(\frac{dV(\phi_{c})}{d\phi}\right)^{2}r^{4}+O(r^{8})\,,$ (17)
$\displaystyle\phi(r)$
$\displaystyle=\phi_{c}+\frac{1}{6}\frac{dV(\phi_{c})}{d\phi}r^{2}+O(r^{4})\,,$
(18)

where $\sigma_{c}$ and $\phi_{c}$ are the values of $\sigma$ and $\phi$ at the
origin, respectively.

(a)  (b)  (c)

Figure 4: The mass $M$ as a function of the parameter $\phi_{c}$. The purple
curve corresponds to the scalaron, and the rest of colored lines for sets of
black holes with fixed values of $\phi_{H}$ (in this case
$\phi_{c}=\phi_{H}$). The colour gradient indicates the size of the horizon
radius $r_{H}$.

Black holes and scalarons share the same asymptotic expansion of the metric
and scalar functions at infinity. As we take $r\to\infty$, if we impose
asymptotical flatness and the scalar field to vanish, then the leading terms
are given by the following expressions

$\displaystyle m(r)$
$\displaystyle=M+\tilde{m}_{1}\frac{\exp{(-2m_{\text{eff}}r)}}{r}+...\,,$ (19)
$\displaystyle\sigma(r)$
$\displaystyle=\tilde{\sigma}_{1}\frac{\exp{(-2m_{\text{eff}}r)}}{r}+...\,,$
(20) $\displaystyle\phi(r)$
$\displaystyle=\bar{\phi}_{H,1}\frac{\exp{(-m_{\text{eff}}r)}}{r}+...\,,$ (21)

where $\tilde{m}_{1}$, $\tilde{\sigma}_{1}$ and $\tilde{\phi}_{H,1}$ are
constants, $M$ is the total mass of the configuration. Note that the effective
mass of scalar field is given by
$m_{\text{eff}}=\sqrt{V_{0}\psi_{0}\psi_{1}}$.

We introduce the following dimensionless parameters,

$r\rightarrow\frac{r}{\sqrt{4\pi G}}\,,\quad m\rightarrow\frac{m}{\sqrt{4\pi
G}}\,,\quad\phi\rightarrow\sqrt{4\pi G}\phi\,.$ (22)

The ODEs are solved by an ODE solver package Colsys which employing the
Newton-Raphson method to solve the boundary value problem for a set of
nonlinear ODEs by providing the adaptive mesh refinement to generate the
solutions to have more than 1000 points with high accuracy and the estimation
of errors of solutions Ascher:1979iha . We compactify the radial coordinate
$r$ by $r=r_{H}/(1-x)$ for hairy black holes and $r=x/(1-x)$ for scalaron in
the numerics. Here we have 5 parameters
$(\phi_{H},\phi_{0},\phi_{1},V_{0},r_{H})$ to describe the hairy black holes
and 4 parameters $(\phi_{c},\phi_{0},\phi_{1},V_{0})$ to describe the
scalaron. To generate the solutions, we fix the global minimum $\phi_{1}$, the
value of $\phi_{0}$ is determined exactly when the boundary conditions are
satisfied with the suitable choices of $\phi_{c}$ and $\phi_{H}$ for scalaron
and hairy black holes with fixed $r_{H}$ and $V_{0}$, respectively.

(a)  (b)

Figure 5: Typical profiles of the functions in the compactified coordinate $x$
for (a) Black holes. (b) Scalaron.

If we consider a hairy black hole, our solutions require a non-vanishing
gradient of the scalar field at the event horizon. Therefore, if we maximally
extend the causal structure over the Einstein-Rosen bridge (left of Fig. 2),
one necessarily sees a cusp of the scalar field (black dashed curve) inside
the horizon. However, it is fair to say that our solutions satisfy the
regularity condition at the event horizon, and hence, there is a natural
extension of the solution beyond the event horizon up to the singularity.
Thus, if we restrict the situation that our solution is formed from
astrophysical processes, e.g., gravitational collapses, we can restrict our
solutions above a timelike hypersurface (blue dashed curve), e.g., that
denotes a star surface. So, the final physically sensible causal structure
becomes right of Fig. 2.

##  III Properties of the Scalar Hairy Black Holes

We are interested with the Hawking temperature $T_{H}$ and area of horizon
$A_{H}$ of these black holes,

$T_{H}=\frac{1}{4\pi}N^{\prime}(r_{H})e^{-\sigma_{H}}\,,\quad A_{H}=4\pi
r^{2}_{H}\,,$ (23)

where $\sigma_{H}$ is the value of function $\sigma$ at the horizon. For the
convenience of comparing our black hole solution with other known solutions,
we introduce the following reduced quantities at the horizon of the black
holes,

$a_{H}=\frac{A_{H}}{16\pi M^{2}}\,,\quad t_{H}=8\pi T_{H}M\,.$ (24)

The Ricci scalar $R$ and Kretschmann scalar $K$ for the black hole spacetime
are given by

$\displaystyle R$
$\displaystyle=-N^{\prime\prime}+\frac{3r\sigma^{\prime}-4}{r}N^{\prime}+\frac{2\left(2rN\sigma^{\prime}-N+1+r^{2}N\sigma^{\prime\prime}-r^{2}N\sigma^{\prime
2}\right)}{r^{2}}\,,$ (25) $\displaystyle K$
$\displaystyle=\left(3\sigma^{\prime}N^{\prime}+2N\sigma^{\prime\prime}-N^{\prime\prime}-2N\sigma^{\prime
2}\right)^{2}+\frac{2}{r^{2}}\left(N^{\prime}-2N\sigma^{\prime}\right)^{2}+\frac{2N^{\prime
2}}{r^{2}}+\frac{4(N-1)^{2}}{r^{4}}\,.$ (26)

With the series expansion of functions at the horizon, the $R$ and $K$ are
finite with the leading order,

$\displaystyle R$
$\displaystyle=-\frac{2m_{1}\left(3r_{H}\sigma_{1}-2\right)}{r^{2}_{H}}+\frac{3\sigma_{1}+4m_{2}}{r_{H}}+O\left((r-r_{H})\right)\,,$
(27) $\displaystyle K$
$\displaystyle=\frac{16m^{2}_{2}}{r^{2}_{H}}-\frac{8\left(-2+6m_{1}\sigma_{1}r_{H}+4m_{1}-3\sigma_{1}r_{H}\right)}{r^{3}_{H}}$
$\displaystyle\quad+\frac{12-32m_{1}+48m^{2}_{1}\sigma_{1}r_{H}+36m^{2}_{1}\sigma^{2}_{1}r^{2}_{H}+32m^{2}_{1}+9\sigma^{2}_{1}r^{2}_{H}+12\sigma_{1}r_{H}-36\sigma^{2}_{1}r^{2}_{H}m_{1}-48m_{1}\sigma_{1}r_{H}}{r^{4}_{H}}+O\left((r-r_{H})\right)\,.$
(28)

(a)  (b)

Figure 6: The metric component $g_{rr}$ for a) hairy black hole b) scalaron in
the compactified coordinate $x$.

##  IV The Radial Perturbation

In the case of radial perturbation, we perturb the background metric and
scalar field, respectively as Blazquez-Salcedo:2020nhs

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-N(r)e^{-2\sigma(r)}\left[1+\epsilon e^{-i\omega
t}F_{t}(r)\right]dt^{2}+\frac{1}{N(r)}\left[1+\epsilon e^{-i\omega
t}F_{r}(r)\right]dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\,,$ (29) $\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\phi(r)+\epsilon\Phi_{1}(r)e^{-i\omega t}\,,$ (30)

where $F_{t}(r)$, $F_{r}(r)$ and $\Phi_{1}(r)$ are the small perturbations to
the non-perturbed solutions.

By substituting the above Ansatz to the Einstein equation and KG equation, we
obtain a set of ODEs for the perturbation functions,

$\displaystyle F_{r}$ $\displaystyle=$ $\displaystyle 8\pi
Gr\Phi_{1}\phi^{\prime}\,,$ (31) $\displaystyle F^{\prime}_{t}$
$\displaystyle=$ $\displaystyle-F^{\prime}_{r}+16\pi
Gr\Phi^{\prime}_{1}\phi^{\prime}\,,$ (32)
$\displaystyle\Phi^{\prime\prime}_{1}$ $\displaystyle=$
$\displaystyle\left(\sigma^{\prime}-\frac{N^{\prime}}{N}-\frac{2}{r}\right)\Phi^{\prime}_{1}+\left(\frac{1}{N}\frac{d^{2}V}{d\phi^{2}}-\omega^{2}\frac{e^{2\sigma}}{N^{2}}\right)\Phi_{1}+\frac{F_{r}}{N}\frac{dV}{d\phi}+\frac{F^{\prime}_{r}-F^{\prime}_{t}}{2}\phi^{\prime}\,.$
(33)

We observe that only Eq. (33) is independent because other ODEs (Eqs. (31) and
(32)) are dependent. Hence, we transform $\Phi^{\prime\prime}_{1}$ to a
Schrödinger-like master equation by defining $Z(r)=r\Phi_{1}(r)$,

$\frac{d^{2}Z}{dr_{*}^{2}}+\left(\omega^{2}-V_{R}(r)\right)Z=0\,,$ (34)

with the effective potential $V_{R}(r)$,

$V_{R}(r)=Ne^{-2\sigma}\left[\frac{N}{r}\left(\frac{N^{\prime}}{N}-\sigma^{\prime}\right)-8\pi
GrN\phi^{\prime
2}\left(\frac{N^{\prime}}{N}+\frac{1}{r}-\sigma^{\prime}\right)+16\pi
Gr\phi^{\prime}\frac{dV}{d\phi}+\frac{d^{2}V}{d\phi^{2}}\right]\,.$ (35)

The tortoise coordinate $r_{*}$ is

$\frac{dr_{*}}{dr}=\frac{e^{\sigma}}{N}\,.$ (36)

The perturbation $Z$ is unstable when $\omega^{2}<0$ where the perturbation
grows exponentially with time. In the compactified coordinate $x$, the
effective potential $V_{R}$ has the following expansions at the origin and
horizon, respectively,

$\displaystyle\text{Scalaron}:V_{R}(0)$
$\displaystyle=\tilde{V}_{0}+\tilde{V}_{1}x+\tilde{V}_{2}x^{2}+O(x^{3})\,,$
(37) $\displaystyle\text{Black Hole}:V_{R}(0)$
$\displaystyle=\hat{V}_{1}x+\hat{V}_{2}x^{2}+O(x^{3})\,,$ (38)

where $\tilde{V}_{i}$ and $\hat{V}_{i}$ are constants that depend on the
parameters of the background solution.

Note that Eq. (34) is an eigenvalue problem, hence we compute the radial mode
numerically by using COLSYS to solve it with $\omega^{2}$ as the eigenvalue.
For black holes, we impose that the first-order derivative of the perturbation
function vanishes at the boundaries,
$\partial_{r}Z(r_{H})=\partial_{r}Z(\infty)=0$. In the case of the scalaron,
we impose that the perturbation function vanishes at the boundaries. In the
numerics, we introduce an auxiliary equation $\frac{d}{dr}\omega^{2}=0$, that
allows us to impose an additional condition $Z(r_{p})=1$ at some point
$r_{p}$, which typically lies in the middle of the horizon/origin and
infinity. This allows us to obtain a nontrivial and normalizable solution for
$Z$, since Eq. (34) is homogeneous. The eigenvalue $\omega^{2}$ is found
automatically when $Z$ satisfies all the asymptotic boundary conditions.

(a)  (b)

Figure 7: Profiles of scalar field $\phi$ for a) hairy black hole b) scalaron
in the compactified coordinate $x$.

##  V Results and Discussions

###  V.1 General properties and domain of existence.

By fixing several values of global minimum $\phi_{1}$, we exhibit the
properties of hairy black holes which are the reduced area of horizon $a_{H}$
and reduced Hawking temperature $t_{H}$ in Fig. 3. The purpose to introduce
the reduced quantities is to compare our hairy black holes with the known
solution which is the Schwarzschild black hole in this case. Recall that both
$a_{H}$ and $t_{H}$ are unity for Schwarzschild black hole. By increasing the
value of scalar field at the horizon $\phi_{H}$ from zero, a branch of hairy
black hole solutions emerges from the Schwarzschild black hole. As $\phi_{H}$
increases, $a_{H}$ decreases from unity and $t_{H}$ increases from unity. When
$\phi_{H}\rightarrow\phi_{1}$, $a_{H}$ decreases to zero and $t_{H}$ increases
very sharply for $\phi_{1}=0.5,1.0$. We couldn’t generate configurations for
$\phi_{H}=\phi_{1}$, the solutions becoming singular as we reach this value of
the parameter. Note that when $\phi_{H}=\phi_{1}$, the scalar field sits
exactly at the true vacuum $\phi_{1}$ of the potential and the tunneling
effect does not occur.

For higher values of $\phi_{1}$ (i.e. $\phi_{1}=2.0,3.0,5.0$), the reduced
area $a_{H}$ also decreases almost to zero but the reduced temperature $t_{H}$
remains finite. Again, the solutions should become sick in the limit
$\phi_{H}=\phi_{1}$, but numerically reaching this limit is more complicated,
since the code stopping to work for values of $\phi_{H}$ slighlty below
$\phi_{1}$ (for $\phi_{1}=2.0,3.0$) and $\phi_{H}$ far below $\phi_{1}$ (for
$\phi_{1}=5.0$).

(a)  (b)

Figure 8: (a) The Ricci scalar and (b) The Kretschmann scalar for a hairy
black hole in the compactified coordinate $x$.

In Fig. 4 we show the relation between the mass of hairy black holes and
scalarons with the value of $\phi_{c}$. The scalaron solutions (purple curve)
emerge from the Minkowski spacetime, where the mass is zero and the scalar
field vanishes. As the value of scalar field at the origin $\phi_{c}$
increases from zero, the mass of scalaron also increases. When
$\phi_{c}\rightarrow\phi_{1}$, the mass of scalaron increases very sharply.

The colour bars in the Fig. 4 represent the size of the black hole horizon,
which indicate that the mass of scalaron are connected with hairy black holes
in the small horizon limit $(r_{H}\rightarrow 0)$. Thus, the hairy black holes
possess the scalaron as the limiting configuration in the small horizon limit.
Analogous to the hairy black holes, for $\phi_{1}=0.5,1.0$, the scalarons do
not exist when $\phi_{c}=\phi_{1}$ since the scalar field $\phi_{c}$ sits
exactly at the true vacuum $\phi_{1}$. For large $\phi_{1}$, e.g.,
$\phi_{1}=5.0$, the scalaron solutions are more difficult to generate
numerically, and we cannot reach values of $\phi_{c}$ too close to $\phi_{1}$.

We exhibit the typical profiles of solution in the compactified coordinate $x$
for the hairy black holes and scalaron in Fig 5. Both compact objects show the
similar pattern for the functions. As the value of the scalar field at the
horizon or the origin becomes closer to $\phi_{1}$, these solutions become
closer to bubbles of true vacuum surrounded by the false vacuum. We observe
the solutions have almost constant functions in the bulk, corresponding to
(almost) the global minimum of the potential, and thus the true vacuum
$\phi_{1}$. Moving away from the horizon, the solutions develop a sharp
boundary at some intermediate region of the spacetime, where the functions
rapidly change to another set of almost constant functions. These region
corresponds to the imposed false vacuum $(a=0)$ at infinity, where the scalar
sits in the local minimum.

Moreover, the mass function $m(x)$ possesses a global minimum and can be
strictly non-positive (see Fig 5(b)), which indicates the violation of the
energy conditions. We observe that the global minimum of $m(x)$ decreases very
sharply as $\phi_{H}$ increases to the limit value. This sharp behaviour of
the $m(r)$ function propagates into the metric component $g_{rr}$, that we
display in Fig. 6. The global minimum of $m(x)$ gives rise to the global
maximum of $N(x)$, since $N(x)=1-2m(x)(1-x)/r_{H}$. Thus, the global maximum
of $N(x)$ also gives rise to the global minimum of $g_{rr}$ since
$g_{rr}=1/N(x)$.

To further illustrate how in the limit the spacetime is divided in two
different regions dominated by the two possible vacua, in Fig. 7 we show the
profiles of scalar field for hairy black holes and scalarons. As $\phi_{H}$
and $\phi_{c}$ increase to $\phi_{1}$, the scalar field profile becomes closer
and closer to a step function: the bulk region possess the value of the true
vacuum $\phi_{1}$, while the exterior region possess the value of the false
vacuum $(a=0)$.

The Ricci scalar $R$ and Kretschmann scalar $K$ for a hairy black hole with
$\phi_{1}=1.0$ are shown in Fig. 8(a) and (b), respectively. The Ricci scalar
at the horizon decreases from zero to some negative values when $\phi_{H}$
increases. This indicates that the bulk of hairy black hole possesses negative
curvature. From the horizon to a point where the functions have a sharp
boundary, the profile of $R$ increases from negative value to zero, then to
positive value and then fall to zero again at the infinity. However, the
Krestschmann scalar is positive at the horizon, and its profile decreases
monotonically to zero at the infinity.

###  V.2 Spherical stability.

In Fig. 9, we show the effective potential $V_{R}$ in the compactified
coordinate for the scalaron and hairy black holes. $V_{R}$ possesses finite
value at the origin for scalaron but $V_{R}$ is zero for the hairy black
holes. In this figure it can be seen that the potential possesses always a
negative region for both types of objects, indicating the existence of a
instability.

In fact it is possible to obtain unstable radial modes for all the solutions
we have studied. In Fig. 10 we exhibit the spectra of unstable radial modes
for hairy black holes and scalarons as a function of the $\phi_{c}$ parameter.
The unstable modes decreases from zero to a minimum value as this parameter is
increased. From the minimum, the unstable mode increases again as $\phi_{c}$
approaches the value of $\phi_{1}$ (the numerical results indicate that the
mode becomes zero exactly at this value, not before). The modes for hairy
black holes and scalarons are also smoothly connected in the small horizon
limit. Interestingly we find that the unstable mode of the hairy black holes
decreases in magnitude with the size of the horizon. This indicates that black
holes with large horizon size could be effectively stable (at least in some
time scale).

(a)  (b)

Figure 9: Typical profile of the effective potential $V_{R}(x)$ in the
compactified coordinate $x$ for (a) Scalaron (b) Black holes. In colour we
plot the potential for different values of the $\phi_{c}$ parameter.

##  VI Conclusion

The gravity minimally coupled to a scalar potential allows the construction of
hairy black holes by requiring the scalar potential to be non-strictly
positive for violating the energy conditions Herdeiro:2015waa . Ref.
Corichi:2005pa  employing such scalar potential which has a global minimum
$\phi_{1}$, a local minimum $a$ and a local maximum $\phi_{0}$ to construct
the spherically symmetric and asymptotically flat hairy black holes and mainly
study their empirical mass formulae. The asymptotically flatness condition of
black holes is guaranteed by fixing the local minimum of potential to be zero.
This potential has been widely applied to study the quantum tunneling process
from the false vacuum $a$ to the true vacuum $\phi_{1}$ in the cosmology.

In our paper, we perform a comprehensive study on the properties of black
holes by solving the Einstein-matter field equations numerically. We fix the
global minimum (true vacuum) and vary the value of scalar field at the horizon
$\phi_{H}$ to generate the hairy black holes solutions. Thus, a branch of
hairy black holes with fixed horizon size emerge from the Schwarzshild black
holes. For small $\phi_{1}$, when $\phi_{H}$ increases from zero and
approaches to $\phi_{1}$, the scaled area of horizon decreases from unity to
zero and the scaled Hawking temperature increases very sharply from unity. In
the limit $\phi_{H}=\phi_{1}$, the scalar field sits exactly at the true
vacuum $\phi_{1}$ and no tunneling occurs, thus hairy black holes do not exist
anymore in that limit. For large $\phi_{1}$, the scaled area of horizon also
decreases from unity to zero but the scaled Hawking temperature increases to a
finite value from unity when $\phi_{H}$ increases to a value which is still
less than $\phi_{1}$. In that situation, we are unable to generate the
solutions for $\phi_{H}$ beyond that value because the numerical code stop
working.

Analogous to the hairy black holes, a branch of globally regular particle-like
solution which is known as scalaron emerges from the Minkowski spacetime by
varying the scalar field at the origin $\phi_{c}$. The scalaron also behaves
analogously with the hairy black holes in the limit $\phi_{c}=\phi_{1}$ where
the mass increases very sharply. Similarly, we are also unable to generate the
scalaron for large $\phi_{1}$. In addition, the hairy black holes are reduced
to the scalaron in the small horizon limit.

(a)  (b)  (c)

Figure 10:  The eigenvalue $\omega^{2}$ as a function of the parameter
$\phi_{c}$. The purple curve corresponds to the scalaron, and the rest of
colored lines for sets of black holes with fixed values of $\phi_{H}$ (in this
case $\phi_{c}=\phi_{H}$). The colour gradient indicates the size of the
horizon radius $r_{H}$.

The profiles of both compact objects behave similarly, they have almost the
constant functions in the bulk, which corresponding to the true vacuum
$\phi_{1}$, then have a sharp boundary where the functions rapidly change to
another set of constant functions which corresponding to the false vacuum
$(a=0)$ lies at the infinity.

We also investigate the linear stability of hairy black holes by performing
the radial perturbation on the metric and scalar field. Hence, we obtain a
master equation which is Schrödinger-like. We numerically solve the master
equation to compute the spectra of radial modes. Generically both hairy black
holes and scalaron are unstable against the radial perturbation. Both spectra
decreases from zero to a minimum value and then increases to zero as the value
of scalar field increases from zero and then approaches $\phi_{1}$. Moreover,
the unstable modes of hairy black holes are connected with scalaron in the
small horizon limit. We also find that the hairy black holes with larger
horizon size are more relatively stable against the perturbation.

There are many possible directions that can be derived from this paper. First,
we can study the properties of non-asymptotically flat hairy black holes by
not fixing the local minimum of potential as zero. We also can repeat the same
approach to study the properties of hairy black holes by considering the non-
minimally coupled scalar field with the Ricci scalar Nucamendi:1995ex . Others
are we can consider to study the properties of charged hairy black holes for
this model and their linear stability. Since the astrophysical black holes are
rotating, then it is interesting to study the difference between properties of
rotating hairy black holes with Kerr black hole.

## Acknowledgement

D.Y and XYC are supported by the National Research Foundation of Korea (Grant
No.: 2021R1C1C1008622, 2021R1A4A5031460). XYC is grateful on the hospitality
from the organizer at APCTP in Pohang to present this work in the workshop
String theory, Gravity and Cosmology (SGC2022). JLBS gratefully acknowledges
support from MICINN project PID2021-125617NB-I00, Santander-UCM project
PR44/21‐29910, DFG Research Training Group 1620 Models of Gravity and FCT
project PTDC/FIS-AST/3041/2020. We are grateful to have a useful discussion
with Jutta Kunz.

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