Properties of Scalar Hairy Black Holes and Scalarons with Asymmetric Potential Xiao Yan Chew1 2Dong-han Yeom1 2yand Jose Luis Bl azquez-Salcedo3z 1Department of Physics Education Pusan National University Busan 46241 Republic of Korea

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Properties of Scalar Hairy Black Holes and Scalarons with Asymmetric Potential

Xiao Yan Chew,

1, 2, ∗

Dong-han Yeom,

1, 2, †

and Jose Luis Bl´azquez-Salcedo

3, ‡

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

Departamento de F´sica Te´orica and IPARCOS,

Universidad Complutense de Madrid, E-28040 Madrid, Spain

In this paper we study the properties of black holes and scalarons in Einstein gravity when it is

minimally coupled to a scalar ﬁeld

with an asymmetric potential

(

), constructed in [Phys. Rev.

(2006), 084002] a few decades ago.

(

) 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

. 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 [

] had led

to the construction of non-Abelian hairy black holes [

–

] 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 [

]. In

[

] 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 [

] and later also generalize their model to non-minimally coupled scalar ﬁeld with gravity

[

]. 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 [

–

] 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

(4)-symmetric metric ansatz, the Coleman-De

Luccia instantons explain such a tunneling process via nucleation of a bubble [

]. Even we extend to the spherical

symmetry, again, one can build a solution that explains a tunneling process [

]. 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 [

–

]; also,

some interactions between bubbles may be a source of gravitational waves [27–29].

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 [30,31]. 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

∗Electronic address: xychew998@gmail.com

†Electronic address: innocent.yeom@gmail.com

‡Electronic address: jlblaz01@ucm.es

arXiv:2210.01313v1 [gr-qc] 4 Oct 2022

V(𝜙)

𝜙

𝜙=𝜙0

𝜙=𝜙1

FIG. 1: An illustration of a generic asymmetric scalar potential V(φ).

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

A. Theory and Ans¨atze

The action for Einstein gravity minimally couples with an asymmetric potential

(

) of a scalar ﬁeld

is given by

[8]

S=Zd4x√−gR

16πG −1

2∇µφ∇µφ−V(φ),(1)

where

V(φ) = V0

12 (φ−a)2h3 (φ−a)2−4(φ−a)(φ0+φ1)+6φ0φ1i,(2)

with

φ0

and

φ1

are the constants. As shown in Fig 1, the appearance of cubic term

φ3

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

is the local minimum of potential,

φ0

is the local maximum of potential and

φ1

is the global minimum of potential. Note that 0

φ0< φ1

. In cosmology, this potential can be used to explain a

quantum tunneling process from the false vacuum

to the true vacuum

φ1

. In this paper, we choose

= 0 such that

the hairy black holes is asymptotically ﬂat at the spatial inﬁnity. However, one could obtain a Schwarzschild-AdS like

solutions if ais non-zero [8].

The variation of the action with respect to the metric and scalar ﬁeld yields the Einstein equation and Klein-Gordon

(KG) equation, respectively

Rµν −1

2gµν R= 8πGTµν ,∇µ∇µφ=dV (φ)

dφ ,(3)

where the stress-energy tensor Tµν is given by

Tµν =−gµν 1

2∇αφ∇αφ+V(φ)+∇µφ∇νφ . (4)

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

ds2=−N(r)e−2σ(r)dt2+dr2

N(r)+r2dθ2+ sin2θdϕ2,(5)

where

(

) = 1

−

(

)

with

(

) is the Misner-Sharp mass function. Note that

(

∞

) =

, the total mass of the

conﬁguration.

FIG. 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.

B. Justiﬁcation on the Existence of Hairy Black Holes

Here we brieﬂy justify the existence of hairy black holes by referring to [

]. We multiply

to the Klein-Gordon

equation and integrate it from the black hole horizon to inﬁnity,

Z∞

d4x√−gφ∇µ∇µφ−φdV (φ)

dφ = 0 .(6)

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

Z∞

d4x√−g−∇µφ∇µφ−φdV (φ)

dφ +ZH

d3σnµφ∇µφ= 0 ,(7)

where

nµ

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

nµ∇µφ

= 0 and demand

the scalar ﬁeld falls oﬀ at inﬁnity. Hence, we are left with

Z∞

d4x√−g∇µφ∇µφ+φdV

dφ = 0 ,(8)

Here

∇µφ∇µφ≥

0 because the gradient of

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

φdV

dφ ≤

0. In our case, we choose

be always greater than zero, while the potential

(

) 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

dφ

and repeat again the above procedure, obtaining

Z∞

d4x√−g"d2V

dφ2∇µφ∇µφ+dV

dφ 2#= 0 .(9)

In order to make the terms in the square bracket to vanish non-trivially, it is clear that

d2V

dφ2<

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

possesses a global

minimum with V < 0 in some regions of φ,

ρ=−Tt

t=N

2φ02+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≤

0 in some

region, but for this analysis it is necessary to introduce the metric Ansatz into the action.

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PropertiesofScalarHairyBlackHolesandScalaronswithAsymmetricPotentialXiaoYanChew,1,2,Dong-hanYeom,1,2,yandJoseLuisBlazquez-Salcedo3,z1DepartmentofPhysicsEducation,PusanNationalUniversity,Busan46241,RepublicofKorea2ResearchCenterforDielectricandAdvancedMatterPhysics,PusanNationalUniversity,Busan4624...

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Properties of Scalar Hairy Black Holes and Scalarons with Asymmetric Potential Xiao Yan Chew1 2Dong-han Yeom1 2yand Jose Luis Bl azquez-Salcedo3z 1Department of Physics Education Pusan National University Busan 46241 Republic of Korea

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